Recent zbMATH articles in MSC 15Ahttps://www.zbmath.org/atom/cc/15A2021-05-28T16:06:00+00:00WerkzeugOn stationary solutions for Lindblad-type equations.https://www.zbmath.org/1459.150172021-05-28T16:06:00+00:00"Kilin, S. Ya."https://www.zbmath.org/authors/?q=ai:kilin.sergei-ya"Makarov, E. K."https://www.zbmath.org/authors/?q=ai:makarov.evgenii-konstantinovichSummary: An explicit solution of the matrix equation defining the stationary solutions of the kinetic Lindblad type equations is obtained.New concavity and convexity results for symmetric polynomials and their ratios.https://www.zbmath.org/1459.260272021-05-28T16:06:00+00:00"Sra, Suvrit"https://www.zbmath.org/authors/?q=ai:sra.suvritThe \(p\)-power (\(p\in(0,\infty)\)) of a vector \(\mathbf{x}\in\mathbb{R}_{+}^{n}\) is defined as \(\mathbf{x}^{p}\) \(\equiv(x_{1}^{p},...,x_{n}^{p}).~\)Let \(e_{k}\) and \(h_{k}\) denote the \(k\)-th elementary symmetric polynomial and the \(k\)-th complete homogeneous symmetric polynomial of \(n\) variables, respectively. Generalizing the well known inequalities of Marcus-Lopes, it is proved that the functions \(\left[ e_{k}\left( \mathbf{x}^{p}\right) \right] ^{1/p}/\left[ e_{k-l}\left( \mathbf{x}^{p}\right) \right] ^{1/lp}\) and \(\left[ e_{k}\left( \mathbf{x}^{p}\right) \right] ^{1/pk}\) are concave on \(\mathbb{R}_{+}^{n},\) whenever \(p\in(0,1)\) and \(1\leq l\leq k\leq n.\) The convexity inequalities of McLeod and Baston for complete homogeneous symmetric polynomials are also extended by showing the subadditivity of the functions \(\left[ h_{k}\left( \mathbf{x}^{p}\right) \right] ^{1/pk}\) and \(\left[ h_{k}\left( \mathbf{x}^{p}\right) \right] ^{1/pk}/\left[ h_{1}\left( \mathbf{x}^{p}\right) \right] ^{1/p(k-1)}\) on \(\mathbb{R}_{+}^{n},\) for all parameters \(p\in\lbrack1,\infty)\) and \(k\in\{1,2,3,...\}.\)
Reviewer: Constantin Niculescu (Craiova)On the projection-based commuting solutions of the Yang-Baxter matrix equation.https://www.zbmath.org/1459.150212021-05-28T16:06:00+00:00"Zhou, Duanmei"https://www.zbmath.org/authors/?q=ai:zhou.duanmei"Chen, Guoliang"https://www.zbmath.org/authors/?q=ai:chen.guoliang"Yu, Gaohang"https://www.zbmath.org/authors/?q=ai:yu.gaohang"Zhong, Jian"https://www.zbmath.org/authors/?q=ai:zhong.jianSummary: We study the commuting solutions of the Yang-Baxter matrix equation \(A X A = X A X\) when \(A\) is an arbitrary square matrix. By characterizing its commuting solutions based on projection matrices, we show that projections can be determined by using the generalized eigenspaces corresponding to the eigenvalues of \(A\). Therefore, commuting solutions can be constructed explicitly. Our results are more general than those obtained recently by \textit{Q. Dong} [Appl. Math. Lett. 64, 231--234 (2017; Zbl 1353.15013)],
\textit{J. Ding} and \textit{C. Zhang} [Appl. Math. Lett. 35, 86--89 (2014; Zbl 1314.15011)] and \textit{J. Ding} and \textit{N. H. Rhee} [Appl. Math. Lett. 24, No. 12, 2211--2215 (2011; Zbl 1236.15023)].On certain graded representations of filiform Lie algebras.https://www.zbmath.org/1459.170242021-05-28T16:06:00+00:00"Bernik, Janez"https://www.zbmath.org/authors/?q=ai:bernik.janez"Šivic, Klemen"https://www.zbmath.org/authors/?q=ai:sivic.klemenSummary: Let \(G\subset\mathrm{GL}(V)\) be a connected complex linear algebraic group of the same dimension as \(V\) such that the poset of the Zariski closures of the orbits for its action coincides with a full flag of subspaces of \(V\). Using the classification of graded filiform Lie algebras, we determine the isomorphism types of the unipotent radical \(U\) of \(G\) in case \(G\) is not nilpotent and \(U\) is of maximal class. In particular, if \(\dim (G)=\dim (V)\geq 11\), there are, up to isomorphism, only two such unipotent groups.Legendre decomposition for tensors.https://www.zbmath.org/1459.650552021-05-28T16:06:00+00:00"Sugiyama, Mahito"https://www.zbmath.org/authors/?q=ai:sugiyama.mahito"Nakahara, Hiroyuki"https://www.zbmath.org/authors/?q=ai:nakahara.hiroyuki"Tsuda, Koji"https://www.zbmath.org/authors/?q=ai:tsuda.kojiVariational representations related to quantum Rényi relative entropies.https://www.zbmath.org/1459.940772021-05-28T16:06:00+00:00"Shi, Guanghua"https://www.zbmath.org/authors/?q=ai:shi.guanghuaSummary: In this paper, we focus on variational representations of some matrix symmetric norm functions that are related to the quantum Rényi relative entropy. Concretely, we obtain variational representations of the function \((A,B)\mapsto |||(B^{q/2}K^\ast A^pKB^{q/2})^s|||\) for symmetric norms by using the Hölder inequality and Young inequality. These variational expressions enable us to make the proofs of the convexity/concavity of the trace function \((A,B)\mapsto\operatorname{Tr}(B^{q/2}K^\ast A^p KB^{q/2})^s\) more clear and give a new extension.Nordhaus-Gaddum-type relations for arithmetic-geometric spectral radius and energy.https://www.zbmath.org/1459.051922021-05-28T16:06:00+00:00"Wang, Yajing"https://www.zbmath.org/authors/?q=ai:wang.yajing"Gao, Yubin"https://www.zbmath.org/authors/?q=ai:gao.yubinSummary: Spectral graph theory plays an important role in engineering. Let \(G\) be a simple graph of order \(n\) with vertex set \(V=\left\{ v_1, v_2, \ldots, v_n\right\}\). For \(v_i\in V \), the degree of the vertex \(v_i\), denoted by \(d_i\), is the number of the vertices adjacent to \(v_i\). The arithmetic-geometric adjacency matrix \(A_{a g}\left( G\right)\) of \(G\) is defined as the \(n\times n\) matrix whose \(\left( i, j\right)\) entry is equal to \(\left( \left( d_i + d_j\right)/2 \sqrt{ d_i d_j}\right)\) if the vertices \(v_i\) and \(v_j\) are adjacent and 0 otherwise. The arithmetic-geometric spectral radius and arithmetic-geometric energy of \(G\) are the spectral radius and energy of its arithmetic-geometric adjacency matrix, respectively. In this paper, some new upper bounds on arithmetic-geometric energy are obtained. In addition, we present the Nordhaus-Gaddum-type relations for arithmetic-geometric spectral radius and arithmetic-geometric energy and characterize corresponding extremal graphs.Tensor train spectral method for learning of hidden Markov models (HMM).https://www.zbmath.org/1459.650162021-05-28T16:06:00+00:00"Kuznetsov, Maxim A."https://www.zbmath.org/authors/?q=ai:kuznetsov.maxim-a"Oseledets, Ivan V."https://www.zbmath.org/authors/?q=ai:oseledets.ivan-vSummary: We propose a new algorithm for spectral learning of Hidden Markov Models (HMM). In contrast to the standard approach, we do not estimate the parameters of the HMM directly, but construct an estimate for the joint probability distribution. The idea is based on the representation of a joint probability distribution as an N-th-order tensor with low ranks represented in the \textit{tensor train} (TT) format. Using TT-format, we get an approximation by minimizing the Frobenius distance between the empirical joint probability distribution and tensors with low TT-ranks with core tensors normalization constraints. We propose an algorithm for the solution of the optimization problem that is based on the alternating least squares (ALS) approach and develop its fast version for sparse tensors. The order of the tensor \(d\) is a parameter of our algorithm. We have compared the performance of our algorithm with the existing algorithm by Hsu, Kakade and Zhang proposed in 2009 and found that it is much more robust if the number of hidden states is overestimated.Transversely isotropic tensor closest in Euclidean norm to a given anisotropic elastic modulus tensor.https://www.zbmath.org/1459.740192021-05-28T16:06:00+00:00"Ostrosablin, N. I."https://www.zbmath.org/authors/?q=ai:ostrosablin.n-iSummary: The problem of determining the transversely isotropic tensor closest in Euclidean norm to a given anisotropic elastic modulus tensor is considered. An orthonormal basis in the space of transversely isotropic tensors for any given axis of symmetry was obtained by decomposition of a transversely isotropic tensor in the general coordinate system into one isotropic part, two deviator parts, and one nonor part. The closest transversely isotropic tensor was obtained by projecting the general anisotropy tensor onto this basis. Equations for five coefficients of the transversely isotropic tensor were derived and solved. Three equations describing stationary conditions were obtained for the direction cosines of the axis of rotation (symmetry). Solving these equations yields the absolute minimum distance from the transversely isotropic tensor to a given anisotropic elastic modulus tensor. The transversely isotropic elastic modulus tensor closest to the cubic symmetry tensor was found.Matrices whose group inverses are \(M\)-matrices.https://www.zbmath.org/1459.150072021-05-28T16:06:00+00:00"Kalauch, A."https://www.zbmath.org/authors/?q=ai:kalauch.anke"Lavanya, S."https://www.zbmath.org/authors/?q=ai:lavanya.s"Sivakumar, K. C."https://www.zbmath.org/authors/?q=ai:sivakumar.koratti-chengalrayanLet \(A\in \mathbb{R}^{n\times n}\) be a nonnegative invertible matrix. Then \(A\) is called an inverse \(M\)-matrix if \(A^{-1}\) is an \(M\)-matrix. The authors first review some known results about \(M\)-matrices whose group inverses are \(M\)-matrices. Then they obtain a necessary and sufficient condition for a matrix \(B\) for which \(B-I\) is an inverse \(M\)-matrix. A similar statement for group inverses is shown to be false. They show that the group inverse of an irreducible matrix is irreducible. They also present two characterizations such that the group inverse of a matrix is an \(M\)-matrix possessing the so-called property \(c\). Given a matrix \(A\) such that its group inverse is an \(M\)-matrix, they present a framework in which a rank one perturbation also inherits this property. In addition, they generalize a result of \textit{Y. Chen} et al. [Linear Algebra Appl. 233, 81--97 (1996; Zbl 0841.15005)]. Finally, they find \(2 \times 2\) matrices and symmetric tridiagonal \(3 \times 3\) matrices such that their group inverses are \(M\)-matrices.
Reviewer: Mohammad Sal Moslehian (Mashhad)The spectral spread of Hermitian matrices.https://www.zbmath.org/1459.420442021-05-28T16:06:00+00:00"Massey, Pedro"https://www.zbmath.org/authors/?q=ai:massey.pedro-g"Stojanoff, Demetrio"https://www.zbmath.org/authors/?q=ai:stojanoff.demetrio"Zárate, Sebastián"https://www.zbmath.org/authors/?q=ai:zarate.sebastianSummary: Let \(A\) be an \(n\times n\) complex Hermitian matrix and let \(\lambda(A)=(\lambda_1,\dots,\lambda_n)\in\mathbb{R}^n\) denote the eigenvalues of \(A\), counting multiplicities and arranged in non-increasing order. Motivated by problems arising in the theory of low rank matrix approximation, we study the spectral spread of \(A\), denoted \(\text{Spr}^+(A)\), given by \(\text{Spr}^+(A)=(\lambda_1-\lambda_n, \lambda_2-\lambda_{n-1},\dots,\lambda_k-\lambda_{n-k+1})\in \mathbb{R}^k\), where \(k=[n/2]\) (integer part). The spectral spread is a vector-valued measure of dispersion of the spectrum of \(A\), that allows one to obtain several submajorization inequalities. In the present work we obtain inequalities that are related to Tao's inequality for anti-diagonal blocks of positive semidefinite matrices, Zhan's inequalities for the singular values of differences of positive semidefinite matrices, extremal properties of direct rotations between subspaces, generalized commutators and distances between matrices in the unitary orbit of a Hermitian matrix.Approximate method of variational Bayesian matrix factorization/completion with sparse prior.https://www.zbmath.org/1459.823252021-05-28T16:06:00+00:00"Kawasumi, Ryota"https://www.zbmath.org/authors/?q=ai:kawasumi.ryota"Takeda, Koujin"https://www.zbmath.org/authors/?q=ai:takeda.koujinOn the eigenvalues of spectral gaps of matrix-valued Schrödinger operators.https://www.zbmath.org/1459.650452021-05-28T16:06:00+00:00"Aljawi, Salma"https://www.zbmath.org/authors/?q=ai:aljawi.salma"Marletta, Marco"https://www.zbmath.org/authors/?q=ai:marletta.marcoSummary: This paper presents a method for calculating eigenvalues lying in the gaps of the essential spectrum of matrix-valued Schrödinger operators. The technique of dissipative perturbation allows eigenvalues of interest to move up the real axis in order to achieve approximations free from spectral pollution. Some results of the behaviour of the corresponding eigenvalues are obtained. The effectiveness of this procedure is illustrated by several numerical examples.Computing enclosures for the matrix exponential.https://www.zbmath.org/1459.650562021-05-28T16:06:00+00:00"Frommer, Andreas"https://www.zbmath.org/authors/?q=ai:frommer.andreas"Hashemi, Behnam"https://www.zbmath.org/authors/?q=ai:hashemi.behnamAn iterative algorithm for solving a class of generalized coupled Sylvester-transpose matrix equations over bisymmetric or skew-anti-symmetric matrices.https://www.zbmath.org/1459.650542021-05-28T16:06:00+00:00"Yan, Tongxin"https://www.zbmath.org/authors/?q=ai:yan.tongxin"Ma, Changfeng"https://www.zbmath.org/authors/?q=ai:ma.changfengSummary: This paper presents an iterative algorithm to solve a class of generalized coupled Sylvester-transpose matrix equations over bisymmetric or skew-anti-symmetric matrices. When the matrix equations are consistent, the bisymmetric or skew-anti-symmetric solutions can be obtained within finite iteration steps in the absence of round-off errors for any initial bisymmetric or skew-anti-symmetric matrices by the proposed iterative algorithm. In addition, we can obtain the least norm solution by choosing the special initial matrices. Finally, numerical examples are given to demonstrate the iterative algorithm is quite efficient. The merit of our method is that it is easy to implement.The family of perfect ideals of codimension 3, of type 2 with 5 generators.https://www.zbmath.org/1459.130122021-05-28T16:06:00+00:00"Celikbas, Ela"https://www.zbmath.org/authors/?q=ai:celikbas.ela"Laxmi, Jai"https://www.zbmath.org/authors/?q=ai:laxmi.jai"Kraśkiewicz, Witold"https://www.zbmath.org/authors/?q=ai:kraskiewicz.witold"Weyman, Jerzy"https://www.zbmath.org/authors/?q=ai:weyman.jerzy-mThe notion of linkage was formalized by \textit{C. Peskine} and \textit{L. Szpiro} [Invent. Math. 26, 271--302 (1974; Zbl 0298.14022)]. Two ideals \(I\) and \(J\) in a Cohen-Macaulay local ring \(R\) are linked (write \(I \sim J\)) if there is an \(R\)-regular sequence \(\alpha=(\alpha_1, \cdots, \alpha_g)\) in their intersection such that \(I=((\underline{\alpha}): J)\) and \(J=((\underline{\alpha}): I)\). To turn linkage into an equivalence relation, one considers the linkage class of an ideal \(I\), as the set of all ideals obtained from \(I\) by a finite steps of links. The ideal \(I\) is said to be \textit{licci} if \(I\) is in the linkage class of a complete intersection ideal.
One of the main themes in linkage theory is characterization of licci ideals. This classification has been settled only for ideals of low dimension: Let \(I\) be an ideal of grade at most two, then \textit{R. Apéry} [C. R. Acad. Sci., Paris 220, 271--272 (1945; Zbl 0061.33602)] and \textit{F. Gaeta} [in: Centre Belge Rech. Math., 2ième Colloque Géom. algébrique, Liège du 9 au 12 juin145--183 (1952; Zbl 0048.14601)] have shown that \(I\) is licci if and only if \(I\) is perfect.
\textit{J. Watanabe} [Nagoya Math. J. 50, 227--232 (1973; Zbl 0242.13019)] has demonstrated perfect ideals of grade \(3\) is licci, if \(R/I\) is Gorenstein.
In the paper under review, the authors provide a family of perfect ideals of grade \(3\) which is licci but \(R/I\) is not Gorenstein.
\textit{A. E. Brown} [J. Algebra 105, 308--327 (1987; Zbl 0624.13010)] has shown that for ideals of grade \(3\), type \(2\), minimally generated by \(5\) elements, number of Koszul relations in the first syzygies, \(\lambda\), is either \(0\) or \(1\). She shows that all of these ideals are licci if \(\lambda=1\). Motivated by the result of Brown, the authors define a family of perfect ideals of grade \(3\), type \(2\), minimally generated by \(5\) elements with \(\lambda=0\) which are licci. The authors construct an ideal \(J\) over the coordinate ring of a pencil of \(4 \times 4\) skew-symmetric matrices and a \(4\)-vector, called \(A\). More precisely a \(5\)-generated ideal, called \(J\), in a bigraded polynomial ring with \(16\) variables. Then the authors provide a deformed ideal \(J(t)\) in the polynomial extension ring \(B=A[t]\). \(J(t)\) constructs a family of perfect ideals of grade \(3\), type \(2\) with \(5\) generators which are all licci, with the minimal betti sequence \((2,6,5,1)\).
Another feature of this family is that, the obstruction condition of non-licciness of \textit{C. Huneke} and \textit{B. Ulrich} [Ann. Math. (2) 126, 277--334 (1987; Zbl 0638.13003)] is sharp.
Reviewer: Maral Mostafazadehfard (Rio de Janeiro)Typical ranks in symmetric matrix completion.https://www.zbmath.org/1459.150292021-05-28T16:06:00+00:00"Bernstein, Daniel Irving"https://www.zbmath.org/authors/?q=ai:bernstein.daniel-irving"Blekherman, Grigoriy"https://www.zbmath.org/authors/?q=ai:blekherman.grigoriy"Lee, Kisun"https://www.zbmath.org/authors/?q=ai:lee.kisunSummary: We study the problem of low-rank matrix completion for symmetric matrices. The minimum rank of a completion of a generic partially specified symmetric matrix depends only on the location of the specified entries, and not their values, if complex entries are allowed. When the entries are required to be real, this is no longer the case and the possible minimum ranks are called \textit{typical ranks}. We give a combinatorial description of the patterns of specified entries of \(n\times n\) symmetric matrices that have \(n\) as a typical rank. Moreover, we describe exactly when such a generic partial matrix is minimally completable to rank \(n\). We also characterize the typical ranks for patterns of entries with low maximal typical rank.On the rank of a random binary matrix.https://www.zbmath.org/1459.150372021-05-28T16:06:00+00:00"Cooper, Colin"https://www.zbmath.org/authors/?q=ai:cooper.colin"Frieze, Alan"https://www.zbmath.org/authors/?q=ai:frieze.alan-m"Pegden, Wesley"https://www.zbmath.org/authors/?q=ai:pegden.wesleySummary: We study the rank of a random \(n \times m\) matrix \(\mathbf{A}_{n,m;k}\) with entries from \(GF(2)\), and exactly \(k\) unit entries in each column, the other entries being zero. The columns are chosen independently and uniformly at random from the set of all \(\binom{n}{k}\) such columns.
We obtain an asymptotically correct estimate for the rank as a function of the number of columns \(m\) in terms of \(c,n,k\), and where \(m=cn/k\). The matrix \(\mathbf{A}_{n,m;k}\) forms the vertex-edge incidence matrix of a \(k\)-uniform random hypergraph \(H\). The rank of \(\mathbf{A}_{n,m;k}\) can be expressed as follows. Let \(|C_2|\) be the number of vertices of the 2-core of \(H\), and \(|E(C_2)|\) the number of edges. Let \(m^*\) be the value of \(m\) for which \(|C_2|= |E(C_2)|\). Then w.h.p. for \(m<m^*\) the rank of \(\mathbf{A}_{n,m;k}\) is asymptotic to \(m\), and for \(m \ge m^*\) the rank is asymptotic to \(m-|E(C_2)|+|C_2|\).
In addition, assign i.i.d. \(U[0,1]\) weights \(X_i, i \in{1,2,\ldots m}\) to the columns, and define the weight of a set of columns \(S\) as \(X(S)=\sum_{j \in S} X_j\). Define a basis as a set of \(n-\mathbb{1} (k\text{ even})\) linearly independent columns. We obtain an asymptotically correct estimate for the minimum weight basis. This generalises the well-known result of \textit{A. M. Frieze} [Discrete Appl. Math. 10, 47--56 (1985; Zbl 0578.05015)] that, for \(k=2\), the expected length of a minimum weight spanning tree tends to \(\zeta(3)\sim 1.202\).Stability of gyroscopic systems with respect to perturbations.https://www.zbmath.org/1459.700192021-05-28T16:06:00+00:00"Guglielmi, Nicola"https://www.zbmath.org/authors/?q=ai:guglielmi.nicola"Manetta, Manuela"https://www.zbmath.org/authors/?q=ai:manetta.manuelaSummary: A linear gyroscopic system is of the form: \[M \ddot{x} + G\dot{x} + K x = 0,\] where the mass matrix \(M\) is a symmetric positive definite real matrix, the gyroscopic matrix \(G\) is real and skew symmetric, and the stiffness matrix \(K\) is real and symmetric. The system is stable if and only if the quadratic eigenvalue problem \(\det (\lambda^2 M+\lambda G + K)=0\) has all eigenvalues on the imaginary axis.
In this chapter, we are interested in evaluating robustness of a given stable gyroscopic system with respect to perturbations. In order to do this, we present an ODE-based methodology which aims to compute the closest unstable gyroscopic system with respect to the Frobenius distance.
A few examples illustrate the effectiveness of the methodology.
For the entire collection see [Zbl 1416.65008].Finding a nonnegative solution to an M-tensor equation.https://www.zbmath.org/1459.150182021-05-28T16:06:00+00:00"Li, Dong-Hui"https://www.zbmath.org/authors/?q=ai:li.donghui|li.dong-hui"Guan, Hong-Bo"https://www.zbmath.org/authors/?q=ai:guan.hongbo"Wang, Xiao-Zhou"https://www.zbmath.org/authors/?q=ai:wang.xiaozhouSummary: We are concerned with the tensor equation with an M-tensor, which we call the M-tensor equation. We first derive a necessary and sufficient condition for an M-tensor equation to have nonnegative solutions. We then develop a monotone iterative method to find a nonnegative solution to an M-tensor equation. The method can be regarded as an approximation to Newton's method for solving the equation. At each iteration, we solve a system of linear equations. An advantage of the proposed method is that the coefficient matrices of the linear systems are independent of the iteration. We show that if the initial point is appropriately chosen, then the sequence of iterates generated by the method converges to a nonnegative solution of the M-tensor equation monotonically and linearly. At last, we do numerical experiments to test the proposed methods. The results show the efficiency of the proposed methods.Definite determinantal representations via orthostochastic matrices.https://www.zbmath.org/1459.150282021-05-28T16:06:00+00:00"Dey, Papri"https://www.zbmath.org/authors/?q=ai:dey.papriA necessary and sufficient condition to guarantee the existence of a definite determinantal representation of a bivariate polynomial by identifying its coefficients as scalar products (defined by orthostochastic matrices) of two vectors is obtained and employed for developing a method for computing a monic symmetric/Hermitian determinantal representation of a bivariate polynomial of a given degree. A computational relaxation to the determinantal problem is also investigated, taking into consideration the observation that it can be reformulated as a problem of expressing the vector of coefficients of the given polynomial as convex combinations of some specified points. Some examples complete the theoretical investigations.
Reviewer: Sorin-Mihai Grad (Wien)Sufficient conditions for the solvability of a Sylvester-like absolute value matrix equation.https://www.zbmath.org/1459.150162021-05-28T16:06:00+00:00"Hashemi, Behnam"https://www.zbmath.org/authors/?q=ai:hashemi.behnamThe author considers real matrix equations of the form (*) \(AXB+C\left\vert X\right\vert D=E\), where \(A,B,C,D\) and \(E\) are given rectangular matrices of appropriate sizes and \(X\) is the unknown matrix. The absolute value of \(X\) is denoted by \(\left\vert X\right\vert \) and inequalities such as \(X\leq Y\) should be interpreted component-wise. The special case (**) \(Ax-C\left\vert x\right\vert =e\), where \(e\) and \(x\) are column matrices is closely related to a problem in linear programming. Conditions for the existence and the uniqueness of solutions for (**) have been studied in a series of papers, beginning with [\textit{J. Rohn}, Linear Algebra Appl. 126, 39--78 (1989; Zbl 0712.65029)]. In the present paper, the author shows how these results can be extended to the general case (*) using Kronecker products.
If \(A\) is an \(m\times n\) matrix then \(\mathrm{vec}(A)\) is the vector \(mn\)-vector obtained by stacking the successive columns of \(A\) below one another. It is well known that for compatible matrices \(\mathrm{vec} (ABC)=(C^{T}\otimes A)\mathrm{vec}(B)\) (see, for example, [\textit{R. A. Horn} and \textit{C. R. Johnson}, Matrix analysis. 2nd ed. Cambridge: Cambridge University Press (2013; Zbl 1267.15001)]). Using this observation, the equation (*) can be rewritten as an equation of the form (**) in which \(x:=\mathrm{vec}(X)\). The following theorems are then proved.
Theorem 1. The equation (*) has a unique solution for each \(E\) if any of the following is true:
(i) the singular values satisfy \(\sigma_{\max} (C)\sigma_{\max}(D)<\sigma_{\min}(A)\sigma_{\min}(B)\);
(ii) \(A\) and \(B\) are square and nonsingular and the spectral radii satisfy \[\rho(|A^{-1} C|)\rho(|B^{-1}D|)<1;\]
(iii) \(A\) and \(B\) are square and the only solution to \(|AXB|\leq|C||X||D|\) is \(X=0\).
Theorem 2. If \(C\) is square and nonsingular with \(0\neq C^{-1}E\geq0\) and \(\sigma_{\max}(A)\sigma _{\max}(B)<\sigma_{\min}(C)\), then \(AXB-C|X|=E\) has no solution.
Reviewer: John D. Dixon (Ottawa)Cholesky decomposition of matrices over commutative semirings.https://www.zbmath.org/1459.160432021-05-28T16:06:00+00:00"Dolžan, David"https://www.zbmath.org/authors/?q=ai:dolzan.david"Oblak, Polona"https://www.zbmath.org/authors/?q=ai:oblak.polonaSummary: We prove that over a commutative semiring every symmetric strongly invertible matrix with nonnegative numerical range has a Cholesky decomposition.Higher order normal modes.https://www.zbmath.org/1459.370412021-05-28T16:06:00+00:00"Gaeta, Giuseppe"https://www.zbmath.org/authors/?q=ai:gaeta.giuseppe"Walcher, Sebastian"https://www.zbmath.org/authors/?q=ai:walcher.sebastianThe authors study Hamiltonian systems with Hamiltonian given by the sum of a classical kinetic energy term (a quadratic function of the momentum) and a potential energy term which is a homogeneous function of the position. Solutions to the equation of motion are found. They are in the form of normal modes, namely the position vector remains on a fixed line passing through the origin. The example of a potential energy which is homogenous of order four is studied in detail.
Reviewer: Mohammad Khorrami (Tehran)The Sherman-Morrison-Woodbury formula for the Moore-Penrose metric generalized inverse.https://www.zbmath.org/1459.150082021-05-28T16:06:00+00:00"Shi, Dongwei"https://www.zbmath.org/authors/?q=ai:shi.dongwei"Cao, Jianbing"https://www.zbmath.org/authors/?q=ai:cao.jianbingLet \(X\) and \(Y\) be Banach spaces. We say that a map \(A:X\rightarrow Y\) is quasi-additive on a subset \(M\subseteq X\) if
\[
A(x+z)=A(x)+A(z)\quad \text{for every pair }x\in X\text{ and }z\in M.
\]
A map \(A:X\rightarrow Y\) is said to be a bounded homogeneous operator when \(A\) maps every bounded set in \(X\) into a bounded set \(Y\) and \(A(\lambda x)=\lambda A(x)\) for every \(x\in X\) and every \(\lambda \in \mathbb{R}\). The set of all bounded homogeneous operators from \(X\) to \(X\) is denoted by \(H(X)\). Let \(B(X,Y)\) be the set of all bounded linear operators from \(X\) to \(Y\).
When \(X=Y\) we write \(B(X)\) instead of \(B(X,X)\). We denote by \(\mathcal{N}(A)\) and \(\mathcal{R}(A)\) the null space and the range of \(A\in B(X,Y)\), respectively. Suppose \(\mathcal{N}(A)\) and \(\mathcal{R}(A)\) are Chebyshev subspaces of \(X\) and \(Y\), respectively. The concept of the Moore-Penrose metric generalized inverse of \(A\in B(X,Y)\) is a generalization of the well-known Moore-Penrose inverse for an operator (with a closed range) in \(B(X,Y)\) where \(X\) and \(Y\) are Hilbert spaces. It turns out that when \(X\) and \(Y\) are reflexive and strictly convex Banach spaces, the Moore-Penrose metric generalized inverse of \(A\in B(X,Y)\) uniquely exists if \(\mathcal{R}(A)\) is closed. When such a generalized inverse of \(A\in B(X,Y)\) exists, it is denoted by \(A^{M}\).
From now on, let \(X\) and \(Y\) be reflexive and strictly convex Banach spaces. In this paper, the authors consider a generalization of the well-known Sherman-Morrison-Woodbury formula for matrices in \(B(X,Y)\), where the usual inverse of a matrix is replaced by the Moore-Penrose metric generalized inverse. One of the main result of the paper follows.
Theorem. Let \(A\in B(X)\), \(G\in B(Y)\), \(U\in B(Y,X)\), \(V\in B(X,Y)\), and let \(\mathcal{R}(A)\) and \(\mathcal{R}(G)\) be closed. Let \(B=A+UGV\in B(X)\) and \(C=G^{M}+VA^{M}U\in H(Y)\) be such that \(\mathcal{R}(B)\) and \(\mathcal{R}(C)\) are closed. If \(B^{M}\) is quasi-additive on both \(\mathcal{R}(A)\) and \(\mathcal{R}(U)\), and
\[
\begin{gathered}
\mathcal{R}(A^{M}) \subseteq \mathcal{R}(B^{M}),\quad \mathcal{N}(A^{M}) \subseteq \mathcal{N}(B^{M}), \\
\mathcal{N}(G^{M}) \subseteq \mathcal{N}(U),\quad \mathcal{N}(C^{M}) \subseteq \mathcal{N}(G),
\end{gathered}
\]
then
\[
(A+UGV)^{M}=A^{M}-A^{M}U(G^{M}+VA^{M}U)^{M}VA^{M}.
\]
In addition to the above result, the authors present some conditions for \((A+UGV)^{M}\) to exist, and obtain (as special cases of the main results) generalizations of some known results.
Reviewer: Janko Marovt (Maribor)A generalization of Rohn's theorem on full-rank interval matrices.https://www.zbmath.org/1459.150032021-05-28T16:06:00+00:00"Rubei, Elena"https://www.zbmath.org/authors/?q=ai:rubei.elenaA \(p\times q\) interval matrix is a \(p\times q\) matrix whose entries are closed bounded nonempty intervals in \(\mathbb{R}\). A general (closed) interval matrix is a matrix whose entries are (closed) connected nonempty subsets of \(\mathbb{R}\). Let \(\mu\) be a \(p\times q\) general interval matrix with entries \(\mu _{i,j}\), \(1\leq i\leq p\), \(1\leq j\leq q\). We say that a \(p\times q\) real matrix \(A=(a_{i,j})\) is contained in \(\mu\) if \(a_{i,j}\in \mu _{i,j}\) for any pair \(i,j\). The matrix \(\mu\) has full rank when all matrices contained in \(\mu\) have rank equal to the minimum of \(\{p,q\}\).
\textit{J. Rohn} [Electron. J. Linear Algebra 18, 500--512 (2009; Zbl 1189.65088); Linear Algebra Appl. 126, 39--78 (1989; Zbl 0712.65029); Reliab. Comput. 2, No. 2, 167--171 (1996; Zbl 0855.65037)]
characterized full rank \(p\times q\) with \(p\geq q\) interval matrices.
The main result of this paper is a generalization of Rohn's results [loc. cit.] to general closed interval matrices. In the first part of the paper, the author presents definitions that are required for the statement of the main result, which is quite technical (Theorem 3.6). In the second part of the paper, some preliminary lemmas are first given and then the main theorem is stated and proved. To better illustrate the main result, the author gives three examples of general closed interval matrices where one of them is of full rank (it satisfies the conditions of Theorem 3.6) and two of them are not.
The author concludes the paper with an open problem, i.e., a characterization of full-rank interval matrices whose entries are (not necessarily closed) connected subsets of \(\mathbb{R}\).
Reviewer: Janko Marovt (Maribor)Numerical range and positive block matrices.https://www.zbmath.org/1459.150242021-05-28T16:06:00+00:00"Bourin, Jean-Christophe"https://www.zbmath.org/authors/?q=ai:bourin.jean-christophe"Lee, Eun-Young"https://www.zbmath.org/authors/?q=ai:lee.eun-youngConsider a Hermitian, positive semidefinite \((2n)\times (2n)\) matrix
\[
\begin{pmatrix}
A & X \\ X^* & B
\end{pmatrix},
\]
where \(A\), \(B\) and \(X\) are \(n\times n\) matrices.
Denote by \(W(X)\) the numerical range of \(X\) and by \(d\) the distance of \(W(X)\) to 0.
Let \(\|\cdot\|\) be a symmetric norm on the space \(\mathbb{M}_{2n}\) of complex \((2n)\times (2n)\) matrices, i.e., \(\|UMV\|=\|M\|\) for all \(M\in \mathbb{M}_{2n}\)
and all unitary matrices \(U,V\in \mathbb{M}_{2n}\).
The authors prove the following inequality:
\[
\Bigg\|\begin{pmatrix}A & X\\ X^* & B\end{pmatrix}\Bigg\|\geq\Bigg\|\Bigg(\frac{A+B}{2}+dI\Bigg)\oplus\Bigg(\frac{A+B}{2}-dI\Bigg)\Bigg\|,
\]
where \(I\) is the \(n\times n\) identity matrix and
\[
C\oplus D:=\begin{pmatrix}C & 0 \\ 0 & D\end{pmatrix}
\]
for \(n\times n\) matrices \(C\) and \(D\).
Several corollaries are obtained, for instance the following estimate for the diameter of the numerical range:
\[
\mathrm{diam}\,W\begin{pmatrix}A & X \\ X^* & B\end{pmatrix}-\mathrm{diam}\,W\Bigg(\frac{A+B}{2}\Bigg)\geq 2d
\]
Reviewer: Jan-David Hardtke (Leipzig)An inverse eigenvalue problem for modified pseudo-Jacobi matrices.https://www.zbmath.org/1459.650522021-05-28T16:06:00+00:00"Xu, Wei-Ru"https://www.zbmath.org/authors/?q=ai:xu.weiru"Bebiano, Natália"https://www.zbmath.org/authors/?q=ai:bebiano.natalia"Chen, Guo-Liang"https://www.zbmath.org/authors/?q=ai:chen.guoliangSummary: In this paper, we investigate an inverse eigenvalue problem for matrices that are obtained from pseudo-Jacobi matrices by only modifying the \(( 1 , r )\)-th and \(( r , 1 )\)-th entries, \( 3 \leq r \leq n\). Necessary and sufficient conditions under which the problem is solvable are derived. Uniqueness results are presented and an algorithm to reconstruct the matrices from the given spectral data is proposed. Illustrative examples are provided.Bernoulli F-polynomials and fibo-Bernoulli matrices.https://www.zbmath.org/1459.110642021-05-28T16:06:00+00:00"Kuş, Semra"https://www.zbmath.org/authors/?q=ai:kus.semra"Tuglu, Naim"https://www.zbmath.org/authors/?q=ai:tuglu.naim"Kim, Taekyun"https://www.zbmath.org/authors/?q=ai:kim.taekyunSummary: In this article, we define the Euler-Fibonacci numbers, polynomials and their exponential generating function. Several relations are established involving the Bernoulli F-polynomials, the Euler-Fibonacci numbers and the Euler-Fibonacci polynomials. A new exponential generating function is obtained for the Bernoulli F-polynomials. Also, we describe the Fibo-Bernoulli matrix, the Fibo-Euler matrix and the Fibo-Euler polynomial matrix by using the Bernoulli F-polynomials, the Euler-Fibonacci numbers and the Euler-Fibonacci polynomials, respectively. Factorization of the Fibo-Bernoulli matrix is obtained by using the generalized Fibo-Pascal matrix and a special matrix whose entries are the Bernoulli-Fibonacci numbers. The inverse of the Fibo-Bernoulli matrix is also found.Doubly (sub)stochastic operators on \(\ell^p\) spaces.https://www.zbmath.org/1459.150352021-05-28T16:06:00+00:00"Eshkaftaki, Ali Bayati"https://www.zbmath.org/authors/?q=ai:eshkaftaki.ali-bayatiSummary: A square matrix is said to be doubly stochastic if its elements are non-negative and all row sums and column sums are equal one. An important tool in the study of majorization for infinite dimensional spaces \(\ell^p(I)\) are doubly stochastic operators. These operators are a generalization of doubly stochastic matrices. In this paper, we compare some properties of doubly stochastic operators in finite and infinite dimensions. We will see that if \(D : \ell^p(I) \to \ell^p(I)\) is a doubly stochastic operator then \(\| D \| \leq 1\). Moreover, the existence of such an operator with \(\| D \| < 1\) is equivalent to \(1 < p < \infty\) and \(I\) is an infinite set. We discuss some other properties of doubly stochastic operators such as compactness and closedness and also, provide relevant applications of these operators in the existence of solutions for some infinite linear equations and functional equations.Matrix trace inequalities related to the Tsallis relative entropies of real order.https://www.zbmath.org/1459.810132021-05-28T16:06:00+00:00"Fujii, Masatoshi"https://www.zbmath.org/authors/?q=ai:fujii.masatoshi"Seo, Yuki"https://www.zbmath.org/authors/?q=ai:seo.yukiSummary: In this paper, we show matrix trace inequalities related to the Tsallis relative entropy of real order: For positive definite matrices \(\rho\) and \(\sigma \), and each \(0 < \alpha \leq 1\) \[\operatorname{D}_\alpha(\rho | \sigma) \leq - \operatorname{Tr}\left[ \frac{ \rho^{1 - q}}{ q} T_{\frac{ \alpha}{ q}} ( \rho^q | \sigma^q )\right]\] for all \(q \geq \alpha > 0\), where the Tsallis relative entropy \(\operatorname{D}_\alpha(\rho | \sigma)\) is defined by \(\operatorname{D}_\alpha(\rho | \sigma) = - \operatorname{Tr}(\frac{ \rho^{1 - \alpha} \sigma^\alpha - \rho}{ \alpha})\) and the Tsallis relative operator entropy \(T_\alpha(\rho | \sigma)\) is defined by \(T_\alpha(\rho | \sigma) = \frac{ \rho \sharp_\alpha \sigma - \rho}{ \alpha} \), where \(\sharp_\alpha\) is the matrix \(\alpha \)-geometric mean. Moreover, we show estimates of the difference between two Tsallis relative entropies of real order.Inconsistent LR fuzzy matrix equation.https://www.zbmath.org/1459.150312021-05-28T16:06:00+00:00"Guo, Xiaobin"https://www.zbmath.org/authors/?q=ai:guo.xiaobin"Wu, Lijuan"https://www.zbmath.org/authors/?q=ai:wu.lijuanSummary: In this paper, the inconsistent LR fuzzy matrix equation \(A\widetilde{X}=\widetilde{B}\) is proposed and discussed. Firstly, the LR fuzzy matrix equation is transformed into two crisp matrix equations in which one determines the mean value and the other determines the left and right extends of fuzzy approximate solution. Secondly, the approximate solution of the LR fuzzy matrix equation is obtained by solving two crisp matrix equations according to the generalized inverse of crisp matrix theory. Then, sufficient conditions for the existence of strong LR fuzzy approximate solution are given. Finally, some numerical examples are given to illustrate our proposed method.Positive maps and trace polynomials from the symmetric group.https://www.zbmath.org/1459.810232021-05-28T16:06:00+00:00"Huber, Felix"https://www.zbmath.org/authors/?q=ai:huber.felix-michael|huber.felix-mSummary: With techniques borrowed from quantum information theory, we develop a method to systematically obtain operator inequalities and identities in several matrix variables. These take the form of trace polynomials: polynomial-like expressions that involve matrix monomials \(X_{\alpha_1}, \ldots, X_{\alpha_r}\) and their traces \(\operatorname{tr}(X_{\alpha_1}, \ldots, X_{\alpha_r})\). Our method rests on translating the action of the symmetric group on tensor product spaces into that of matrix multiplication. As a result, we extend the polarized Cayley-Hamilton identity to an operator inequality on the positive cone, characterize the set of multilinear equivariant positive maps in terms of Werner state witnesses, and construct permutation polynomials and tensor polynomial identities on tensor product spaces. We give connections to concepts in quantum information theory and invariant theory.
{\copyright 2021 American Institute of Physics}Eigenvalues and eigenvectors: generalized and determinant free.https://www.zbmath.org/1459.150122021-05-28T16:06:00+00:00"Suzuki, Jeff"https://www.zbmath.org/authors/?q=ai:suzuki.jeffSummary: The standard approach to finding eigenvalues relies on solving the characteristic polynomial. But the characteristic polynomial for an \(n \times n\) matrix will require computing a determinant with \(n!\) terms and solving an \(n\)th degree polynomial equation, both of which are daunting tasks if \(n\geq 3\). We present a determinant-free approach which often leads to lower-degree polynomial equations, and which provides a natural introduction to the concept of a generalized eigenvector.Krylov type methods for linear systems exploiting properties of the quadratic numerical range.https://www.zbmath.org/1459.650412021-05-28T16:06:00+00:00"Frommer, Andreas"https://www.zbmath.org/authors/?q=ai:frommer.andreas"Jacob, Brigit"https://www.zbmath.org/authors/?q=ai:jacob.brigit"Kahl, Kartsen"https://www.zbmath.org/authors/?q=ai:kahl.kartsen"Wyss, Christian"https://www.zbmath.org/authors/?q=ai:wyss.christian"Zwaan, Ian"https://www.zbmath.org/authors/?q=ai:zwaan.ian-nSummary: The quadratic numerical range \(W^2(A)\) is a subset of the standard numerical range of a linear operator, which still contains its spectrum. It arises naturally in operators that have a \(2 \times 2\) block structure, and it consists of at most two connected components, none of which necessarily convex. The quadratic numerical range can thus reveal spectral gaps, and it can in particular indicate that the spectrum of an operator is bounded away from \(0\).
We exploit this property in the finite-dimensional setting to derive Krylov subspace-type methods to solve the system \(Ax = b\), in which the iterates arise as solutions of low-dimensional models of the operator whose quadratic numerical range is contained in \(W^2(A)\). This implies that the iterates are always well-defined and that, as opposed to standard FOM, large variations in the approximation quality of consecutive iterates are avoided, although \(0\) lies within the convex hull of the spectrum. We also consider GMRES variants that are obtained in a similar spirit. We derive theoretical results on basic properties of these methods, review methods on how to compute the required bases in a stable manner, and present results of several numerical experiments illustrating improvements over standard FOM and GMRES.On elliptic biquaternion matrices.https://www.zbmath.org/1459.150332021-05-28T16:06:00+00:00"Yu, Cui-E"https://www.zbmath.org/authors/?q=ai:yu.cuie"Liu, Xin"https://www.zbmath.org/authors/?q=ai:liu.xin|liu.xin.5|liu.xin.2|liu.xin.4|liu.xin.3|liu.xin.1"Zhang, Yang"https://www.zbmath.org/authors/?q=ai:zhang.yang.1|zhang.yang.6|zhang.yang|zhang.yang.5|zhang.yang.2|zhang.yang.3|zhang.yang.7The authors investigate classical problems of linear algebra of matrices (eigenvalues, eigenvectors, singular value decomposition, etc.) for matrices over elliptic biquaternions. They introduce the new concept of the quaternionic adjoint matrix of an elliptic biquaternionic matrix. Among other things, the authors also derive the least-squares solutions of the elliptic biquaternionic matrix equations \(AX = B\), \(XA = B\), and \(AX-XB=C\).
Reviewer: Yansheng Wu (Nanjing)Efficient preconditioning for noisy separable nonnegative matrix factorization problems by successive projection based low-rank approximations.https://www.zbmath.org/1459.650392021-05-28T16:06:00+00:00"Mizutani, Tomohiko"https://www.zbmath.org/authors/?q=ai:mizutani.tomohiko"Tanaka, Mirai"https://www.zbmath.org/authors/?q=ai:tanaka.miraiSummary: The successive projection algorithm (SPA) can quickly solve a nonnegative matrix factorization problem under a separability assumption. Even if noise is added to the problem, SPA is robust as long as the perturbations caused by the noise are small. In particular, robustness against noise should be high when handling the problems arising from real applications. The preconditioner proposed by \textit{N. Gillis} and \textit{S. A. Vavasis} [SIAM J. Optim. 25, No. 1, 677--698 (2015; Zbl 1316.15015)] makes it possible to enhance the noise robustness of SPA. Meanwhile, an additional computational cost is required. The construction of the preconditioner contains a step to compute the top-\(k\) truncated singular value decomposition of an input matrix. It is known that the decomposition provides the best rank-\(k\) approximation to the input matrix; in other words, a matrix with the smallest approximation error among all matrices of rank less than \(k\). This step is an obstacle to an efficient implementation of the preconditioned SPA. To address the cost issue, we propose a modification of the algorithm for constructing the preconditioner. Although the original algorithm uses the best rank-\(k\) approximation, instead of it, our modification uses an alternative. Ideally, this alternative should have high approximation accuracy and low computational cost. To ensure this, our modification employs a rank-\(k\) approximation produced by an SPA based algorithm. We analyze the accuracy of the approximation and evaluate the computational cost of the algorithm. We then present an empirical study revealing the actual performance of the SPA based rank-\(k\) approximation algorithm and the modified preconditioned SPA.The cyclic rank completion problem with general blocks.https://www.zbmath.org/1459.051652021-05-28T16:06:00+00:00"Cohen, Nir"https://www.zbmath.org/authors/?q=ai:cohen.nir"Pereira, Edgar"https://www.zbmath.org/authors/?q=ai:pereira.edgarSummary: We present an upper bound for the minimal completion rank of a partial matrix \(P\) whose block pattern is a single cycle of size \(2k\) with specified blocks \(A_1,\dots,A_{2k}\). Under certain conditions, the bound becomes quite sharp when \(k\) increases. This extends previous results in which the blocks are regular. The upper bound is constructed from invariants associated with the canonical form of the partial matrix, under row and column operations. These invariants can be expressed in terms of ranks of certain matrices constructed directly from the data blocks, independent of \(P\) being in canonical form.On eigenstructure of \(q\)-Bernstein operators.https://www.zbmath.org/1459.650482021-05-28T16:06:00+00:00"Naaz, Ambreen"https://www.zbmath.org/authors/?q=ai:naaz.ambreen"Mursaleen, M."https://www.zbmath.org/authors/?q=ai:mursaleen.mohammad|mursaleen.mohammad-ayman|mursaleen.momammadSummary: The quantum analogue of Bernstein operators \(\mathcal{B}_{m,q}\) reproduce the linear polynomials which are therefore eigenfunctions corresponding to the eigenvalue \(1\), \(\forall q>0\). In this article the rest of eigenstructure of \(q\)-Bernstein operators and the distinct behaviour of zeros of eigenfunctions for cases (i) \(1>q>0\), and (ii) \(q>1\) are discussed. Graphical analysis for some eigenfunctions and their roots are presented with the help of MATLAB. Also, matrix representation for diagonalisation of \(q\)-Bernstein operators is discussed.The signless Laplacian state transfer in coronas.https://www.zbmath.org/1459.051902021-05-28T16:06:00+00:00"Tian, Gui-Xian"https://www.zbmath.org/authors/?q=ai:tian.guixian"Yu, Ping-Kang"https://www.zbmath.org/authors/?q=ai:yu.ping-kang"Cui, Shu-Yu"https://www.zbmath.org/authors/?q=ai:cui.shuyuSummary: For two graphs \(G\) and \(H\), the corona product \(G \circ H\) is the graph obtained by taking one copy of \(G\) and \(|V_G|\) copies of \(H\), and joining the \(i\)th vertex of \(G\) with every vertex of the \(i\)th copy of \(H\). In this paper, we study the state transfer of corona relative to the signless Laplacian matrix. We explore some conditions that guarantee the signless Laplacian perfect state transfer in \(G \circ H\). We prove that \(G \circ K_m\) has no signless Laplacian perfect state transfer for some special \(m\). We also show that \(K_2 \circ H\) has pretty good state transfer but no perfect state transfer relative to the signless Laplacian matrix for a regular graph \(H\). Furthermore, we show that \(\overline{nK_2} \circ K_1\) has signless Laplacian pretty good state transfer, where \(\overline{nK_2}\) is the cocktail party graph.Measuring similarity between connected graphs: the role of induced subgraphs and complementarity eigenvalues.https://www.zbmath.org/1459.051422021-05-28T16:06:00+00:00"Seeger, Alberto"https://www.zbmath.org/authors/?q=ai:seeger.alberto"Sossa, David"https://www.zbmath.org/authors/?q=ai:sossa.davidSummary: This work elaborates on the old problem of measuring the degree of similarity, say \(\mathfrak{f}(G,H)\), between a pair of connected graphs \(G\) and \(H\), not necessarily of the same order. The choice of a similarity index \(\mathfrak{f}\) depends essentially on the graph properties that are considered as important in a given context. As relevant information on a graph, one may consider for instance its degree sequence, its characteristic polynomial, and so on. We explore some new similarity indices based on nonstandard spectral information contained in the graphs under comparison. By nonstandard spectral information in a graph, we mean the set of complementarity eigenvalues of the adjacency matrix. From such a spectral perspective, two distinct graphs \(G\) and \(H\) are viewed as highly similar if they share a large number of complementarity eigenvalues. This basic idea will be cast in a rigorous mathematical formalism.Spectral radii of sparse random matrices.https://www.zbmath.org/1459.150362021-05-28T16:06:00+00:00"Benaych-Georges, Florent"https://www.zbmath.org/authors/?q=ai:benaych-georges.florent"Bordenave, Charles"https://www.zbmath.org/authors/?q=ai:bordenave.charles"Knowles, Antti"https://www.zbmath.org/authors/?q=ai:knowles.anttiThe paper studies spectral radii of classes of random matrices of combinatorial interest, including the adjacency matrices of (inhomogeneous) Erdős-Rényi random graphs. It was already known from [\textit{Z. Füredi} and \textit{J. Komloś}, Combinatorica 1, 233--241 (1981; Zbl 0494.15010); \textit{V. H. Vu}, Combinatorica 27, No. 6, 721--736 (2007; Zbl 1164.05066)] that, for example, in the sparse Erdős-Rényi random graph \(G(n, d/n)\), the second and smallest adjacency eigenvalues converge to the edges of the support of the asymptotic eigenvalue distribution provided \(d/\log(n)^{4}\rightarrow \infty\).
In this paper, these results are extended to a proof that the same statement holds under the weaker assumption that \(d/\log(n)\rightarrow\infty\). A companion paper of the authors shows that in the other regime \(d/\log(n)\rightarrow 0\) the behavior is different [Ann. Probab. 47, No. 3, 1653--1676 (2019; Zbl 1447.60017)].
The main new tool is a refined use of the non-backtracking matrix. It is important to emphasize that the results apply to a much more general class of random graphs, including block stochastic models and inhomogeneous Erdős-Rényi graphs.
Reviewer: David B. Penman (Colchester)Column-oriented algebraic iterative methods for nonnegative constrained least squares problems.https://www.zbmath.org/1459.650432021-05-28T16:06:00+00:00"Nikazad, T."https://www.zbmath.org/authors/?q=ai:nikazad.touraj"Karimpour, M."https://www.zbmath.org/authors/?q=ai:karimpour.mehdiSummary: This paper considers different versions of block-column iterative (BCI) methods for solving nonnegative constrained linear least squares problems. We present the convergence analysis for a family of stationary BCI methods with nonnegativity constraints (BCI-NC), which is applicable to linear complementarity problems (LCP). We also consider the flagging idea for BCI methods, which allows saving computational work by skipping small updates. Also, we combine the BCI-NC algorithm and the flagging version of a nonstationary BCI method with nonnegativity constraints to derive a convergence analysis for the resulting method (BCI-NF). The performance of our algorithms is shown on ill-posed inverse problems taken from tomographic imaging. We compare the BCI-NF and BCI-NC algorithms with three recent algorithms: the inner-outer modulus method (Modulus-CG method), the modulus-based iterative method to Tikhonov regularization with nonnegativity constraint (Mod-TRN method), and nonnegative flexible CGLS (NN-FCGLS) method. Our algorithms are able to produce more stable results than the mentioned methods with competitive computational times.The relationship of inertias between two representations of linear subspaces.https://www.zbmath.org/1459.150042021-05-28T16:06:00+00:00"Ocaña, Eladio"https://www.zbmath.org/authors/?q=ai:ocana.eladio"Flores-Luyo, Luis"https://www.zbmath.org/authors/?q=ai:flores-luyo.luisSummary: This work gives complete expressions of inertia of matrices involved in a linear subspace of \(\mathbb{R}^n\times\mathbb{R}^n\) when they are presented in two different ways, specifically, as image and as well as kernel of linear maps.Identifying influential nodes to enlarge the coupling range of pinning controllability.https://www.zbmath.org/1459.930332021-05-28T16:06:00+00:00"Zhou, Ming-Yang"https://www.zbmath.org/authors/?q=ai:zhou.mingyang"Xu, Rong-Qin"https://www.zbmath.org/authors/?q=ai:xu.rong-qin"Li, Xiao-Yu"https://www.zbmath.org/authors/?q=ai:li.xiaoyu"Liao, Hao"https://www.zbmath.org/authors/?q=ai:liao.haoEigenvalue estimates for multi-form modified Dirac operators.https://www.zbmath.org/1459.353222021-05-28T16:06:00+00:00"Gutowski, Jan"https://www.zbmath.org/authors/?q=ai:gutowski.jan-b"Papadopoulos, George"https://www.zbmath.org/authors/?q=ai:papadopoulos.georgeSummary: We give estimates for the eigenvalues of multi-form modified Dirac operators which are constructed from a standard Dirac operator with the addition of a Clifford algebra element associated to a multi-degree form. In particular such estimates are presented for modified Dirac operators with a \(k\)-degree form \(0\leq k\leq 4\), those modified with multi-degree \((0,k)\)-form \(0\leq k\leq 3\) and the horizon Dirac operators which are modified with a multi-degree \((1,2,4)\)-form. In particular, we give the necessary geometric conditions for such operators to admit zero modes as well as those for the zero modes to be parallel with a respect to a suitable connection. We also demonstrate that manifolds which admit such parallel spinors are associated with twisted covariant form hierarchies which generalize the conformal Killing-Yano forms.On the proportion of transverse-free plane curves.https://www.zbmath.org/1459.140132021-05-28T16:06:00+00:00"Asgarli, Shamil"https://www.zbmath.org/authors/?q=ai:asgarli.shamil"Freidin, Brian"https://www.zbmath.org/authors/?q=ai:freidin.brianSummary: We study the asymptotic proportion of smooth plane curves over a finite field \(\mathbb{F}_q\) which are tangent to every line defined over \(\mathbb{F}_q\). This partially answers a question raised by Charles Favre. Our techniques include applications of Poonen's Bertini theorem and Schrijver's theorem on perfect matchings in regular bipartite graphs. Our main theorem implies that a random smooth plane curve over \(\mathbb{F}_q\) admits a transverse \(\mathbb{F}_q\)-line with very high probability.Symmetric rank-one updates from partial spectrum with an application to out-of-sample extension.https://www.zbmath.org/1459.150102021-05-28T16:06:00+00:00"Mitz, Roy"https://www.zbmath.org/authors/?q=ai:mitz.roy"Sharon, Nir"https://www.zbmath.org/authors/?q=ai:sharon.nir"Shkolnisky, Yoel"https://www.zbmath.org/authors/?q=ai:shkolnisky.yoelNumerical algorithms of the discrete coupled algebraic Riccati equation arising in optimal control systems.https://www.zbmath.org/1459.652372021-05-28T16:06:00+00:00"Wang, Li"https://www.zbmath.org/authors/?q=ai:wang.li.5|wang.li|wang.li.6|wang.li.2|wang.li.1|wang.li.3|wang.li.4Summary: The discrete coupled algebraic Riccati equation (DCARE) has wide applications in robust control, optimal control, and so on. In this paper, we present two iterative algorithms for solving the DCARE. The two iterative algorithms contain both the iterative solution in the last iterative step and the iterative solution in the current iterative step. And, for different initial value, the iterative sequences are increasing and bounded in one algorithm and decreasing and bounded in another. They are all monotonous and convergent. Numerical examples demonstrate the convergence effect of the presented algorithms.A singular value thresholding with diagonal-update algorithm for low-rank matrix completion.https://www.zbmath.org/1459.901512021-05-28T16:06:00+00:00"Duan, Yong-Hong"https://www.zbmath.org/authors/?q=ai:duan.yonghong"Wen, Rui-Ping"https://www.zbmath.org/authors/?q=ai:wen.ruiping"Xiao, Yun"https://www.zbmath.org/authors/?q=ai:xiao.yunSummary: The singular value thresholding (SVT) algorithm plays an important role in the well-known matrix reconstruction problem, and it has many applications in computer vision and recommendation systems. In this paper, an SVT with diagonal-update (D-SVT) algorithm was put forward, which allows the algorithm to make use of simple arithmetic operation and keep the computational cost of each iteration low. The low-rank matrix would be reconstructed well. The convergence of the new algorithm was discussed in detail. Finally, the numerical experiments show the effectiveness of the new algorithm for low-rank matrix completion.Rank-constrained nonnegative matrix factorization for data representation.https://www.zbmath.org/1459.681872021-05-28T16:06:00+00:00"Shu, Zhenqiu"https://www.zbmath.org/authors/?q=ai:shu.zhenqiu"Wu, Xiao-Jun"https://www.zbmath.org/authors/?q=ai:wu.xiaojun"You, Congzhe"https://www.zbmath.org/authors/?q=ai:you.congzhe"Liu, Zhen"https://www.zbmath.org/authors/?q=ai:liu.zhen.1|liu.zhen"Li, Peng"https://www.zbmath.org/authors/?q=ai:li.peng.1|li.peng|li.peng.2|li.peng.4|li.peng.3"Fan, Honghui"https://www.zbmath.org/authors/?q=ai:fan.honghui"Ye, Feiyue"https://www.zbmath.org/authors/?q=ai:ye.feiyueSummary: Graph-based regularized nonnegative matrix factorization (NMF) methods performed well in many real-world applications. However, it is still an open problem to construct an optimal graph to effectively discover the intrinsic geometric structure of data. In this paper, we propose a new data representation framework, called rank-constrained nonnegative matrix factorization (RCNMF). We impose the rank constraint on the Laplacian matrix of the learned graph, so it can ensure that the number of connected components is consistent with the number of sample categories. Instead of a fixed graph-based regularization, the proposed framework can adaptively adjust the weight of the affinity matrix in each iteration. We develop two versions of RCNMF based on the \(l_1\) and \(l_2\) norms, and introduce their optimization schemes. In addition, their convergence and the complexity analyses are also provided. Experimental results on four benchmark datasets show that our methods outperform state-of-the-art methods in clustering.Functions and eigenvectors of partially known matrices with applications to network analysis.https://www.zbmath.org/1459.650462021-05-28T16:06:00+00:00"Al Mugahwi, Mohammed"https://www.zbmath.org/authors/?q=ai:al-mugahwi.mohammed"De la Cruz Cabrera, Omar"https://www.zbmath.org/authors/?q=ai:de-la-cruz-cabrera.omar"Noschese, Silvia"https://www.zbmath.org/authors/?q=ai:noschese.silvia"Reichel, Lothar"https://www.zbmath.org/authors/?q=ai:reichel.lotharSummary: Matrix functions play an important role in applied mathematics. In network analysis, in particular, the exponential of the adjacency matrix associated with a network provides valuable information about connectivity, as well as about the relative importance or centrality of nodes. Another popular approach to rank the nodes of a network is to compute the left Perron vector of the adjacency matrix for the network. The present article addresses the problem of evaluating matrix functions, as well as computing an approximation to the left Perron vector, when only some of the columns and/or some of the rows of the adjacency matrix are known. Applications to network analysis are considered, when only some sampled columns and/or rows of the adjacency matrix that defines the network are available. A sampling scheme that takes the connectivity of the network into account is described. Computed examples illustrate the performance of the methods discussed.A note on the hyperbolic singular value decomposition without hyperexchange matrices.https://www.zbmath.org/1459.150112021-05-28T16:06:00+00:00"Shirokov, D. S."https://www.zbmath.org/authors/?q=ai:shirokov.dmitry-sSummary: We present a new formulation of the hyperbolic singular value decomposition (HSVD) for an arbitrary complex (or real) matrix without hyperexchange matrices and redundant invariant parameters. In our formulation, we use only the concept of pseudo-unitary (or pseudo-orthogonal) matrices. We show that computing the HSVD in the general case is reduced to calculation of eigenvalues, eigenvectors, and generalized eigenvectors of some auxiliary matrices. The new formulation is more natural and useful for some applications. It naturally includes the ordinary singular value decomposition.Least squares solution of the quaternion Sylvester tensor equation.https://www.zbmath.org/1459.150202021-05-28T16:06:00+00:00"Wang, Qing-Wen"https://www.zbmath.org/authors/?q=ai:wang.qingwen"Xu, Xiangjian"https://www.zbmath.org/authors/?q=ai:xu.xiangjian"Duan, Xuefeng"https://www.zbmath.org/authors/?q=ai:duan.xuefengSummary: This paper is concerned with the solution to the least squares problem for the quaternion Sylvester tensor equation. An iterative algorithm based on tensor format is presented to solve this problem. The convergence properties of the proposed iterative method are studied. We also consider the best approximate problem related to quaternion Sylvester tensor equation. Numerical examples are provided to confirm the theoretical results, which demonstrate that the proposed algorithm is effective and feasible for solving the least squares problem of the quaternion Sylvester tensor equation.Copositive and completely positive matrices. 2nd updated and extended edition.https://www.zbmath.org/1459.150022021-05-28T16:06:00+00:00"Shaked-Monderer, Naomi"https://www.zbmath.org/authors/?q=ai:shaked-monderer.naomi"Berman, Abraham"https://www.zbmath.org/authors/?q=ai:berman.abraham-sPublisher's description: This book is an updated and extended version of [the authors, Completely positive matrices. River Edge, NJ: World Scientific (2003; Zbl 1030.15022)]. It contains new sections on the cone of copositive matrices, which is the dual of the cone of completely positive matrices, and new results on both copositive matrices and completely positive matrices.
The book is an up to date comprehensive resource for researchers in matrix theory and optimization. It can also serve as a textbook for an advanced undergraduate or graduate course.Noncommutative invariant theory of symplectic and orthogonal groups.https://www.zbmath.org/1459.053362021-05-28T16:06:00+00:00"Drensky, Vesselin"https://www.zbmath.org/authors/?q=ai:drensky.vesselin"Hristova, Elitza"https://www.zbmath.org/authors/?q=ai:hristova.elitzaLet \(G\) be one of the classical groups \(\mathrm{Sp}_d(\mathbb{C})\) (for \(d\) even), \(\mathrm{O}_d(\mathbb{C})\), or \(\mathrm{SO}_d(\mathbb{C})\). The authors give a method to compute the Hilbert series of the subalgebra of \(G\)-invariants in certain non-commutative graded algebras whose homogeneous components are polynomial \(\mathrm{GL}_d(\mathbb{C})\)-modules. More concretely, let \(W\) be a finite dimensional polynomial \(\mathrm{GL}_d(\mathbb{C})\)-module, \(T(W)\) the tensor algebra of \(W\), \(I\) a \(\mathrm{GL}_d(\mathbb{C})\)-stable ideal in \(T(W)\), and \(H((T(W)/I)^G,z)\) the Hilbert series of the subalgebra \((T(W)/I)^G\) of \(G\)-invariants in \(T(W)/I\). First, a formula is presented that expresses this Hilbert series in terms of the \(\mathrm{GL}_d(\mathbb{C})\)-multiplicity series of \(T(W)/I\). The formula is then applied to compute \(H((T(W)/I)^G,z)\) as an explicit rational function of \(z\) for some \(\mathrm{GL}_d(\mathbb{C})\)-modules \(W\), where \(I=I(\mathfrak{R})\cap T(W)\) is the T-ideal of \(\dim_{\mathbb{C}}(W)\)-variable identities of the variety \(\mathfrak{R}\) of associative algebras generated by the infinite dimensional Grassmann algebra, or the variety \(\mathfrak{R}\) of associative algebras generated by the algebra of \(2\times 2\) upper triangular matrices. In particular, the case when \(W=\mathbb{C}^d\) is the defining \(\mathrm{GL}_d(\mathbb{C})\)-module is settled. Note that the above two varieties are the only minimal varieties of exponent \(2\)
(the exponent is defined in terms of the codimension sequence of \(\mathfrak{R}\)).
Reviewer: Matyas Domokos (Budapest)Tensor \(N\)-tubal rank and its convex relaxation for low-rank tensor recovery.https://www.zbmath.org/1459.681812021-05-28T16:06:00+00:00"Zheng, Yu-Bang"https://www.zbmath.org/authors/?q=ai:zheng.yubang"Huang, Ting-Zhu"https://www.zbmath.org/authors/?q=ai:huang.ting-zhu"Zhao, Xi-Le"https://www.zbmath.org/authors/?q=ai:zhao.xile"Jiang, Tai-Xiang"https://www.zbmath.org/authors/?q=ai:jiang.tai-xiang"Ji, Teng-Yu"https://www.zbmath.org/authors/?q=ai:ji.teng-yu"Ma, Tian-Hui"https://www.zbmath.org/authors/?q=ai:ma.tian-huiSummary: The recent popular tensor tubal rank, defined based on tensor singular value decomposition (t-SVD), yields promising results. However, its framework is applicable only to three-way tensors and lacks the flexibility necessary to handle different correlations along different modes. To tackle these two issues, we define a new tensor unfolding operator, named mode-\( k_1 k_2\) tensor unfolding, as the process of lexicographically stacking all mode-\( k_1 k_2\) slices of an \(N\)-way tensor into a three-way tensor, which is a three-way extension of the well-known mode-\(k\) tensor matricization. On this basis, we define a novel tensor rank, named the tensor \(N\)-tubal rank, as a vector consisting of the tubal ranks of all mode-\( k_1 k_2\) unfolding tensors, to depict the correlations along different modes. To efficiently minimize the proposed \(N\)-tubal rank, we establish its convex relaxation: the weighted sum of the tensor nuclear norm (WSTNN). Then, we apply the WSTNN to low-rank tensor completion (LRTC) and tensor robust principal component analysis (TRPCA). The corresponding WSTNN-based LRTC and TRPCA models are proposed, and two efficient alternating direction method of multipliers (ADMM)-based algorithms are developed to solve the proposed models. Numerical experiments demonstrate that the proposed models significantly outperform the compared ones.Multiview clustering of images with tensor rank minimization via nonconvex approach.https://www.zbmath.org/1459.621252021-05-28T16:06:00+00:00"Yang, Ming"https://www.zbmath.org/authors/?q=ai:yang.ming.1|yang.ming|yang.ming.2"Luo, Qilun"https://www.zbmath.org/authors/?q=ai:luo.qilun"Li, Wen"https://www.zbmath.org/authors/?q=ai:li.wen.1"Xiao, Mingqing"https://www.zbmath.org/authors/?q=ai:xiao.mingqingAnalysis of the Schwarz domain decomposition method for the conductor-like screening continuum model.https://www.zbmath.org/1459.652312021-05-28T16:06:00+00:00"Reusken, Arnold"https://www.zbmath.org/authors/?q=ai:reusken.arnold"Stamm, Benjamin"https://www.zbmath.org/authors/?q=ai:stamm.benjaminWeighted pseudo core inverses in rings.https://www.zbmath.org/1459.160372021-05-28T16:06:00+00:00"Zhu, Huihui"https://www.zbmath.org/authors/?q=ai:zhu.huihui"Wang, Qing-Wen"https://www.zbmath.org/authors/?q=ai:wang.qingwenSummary: Let \(R\) be a unital \(\ast\)-ring and let \(a,e,f\in R\) with \(e,f\) invertible Hermitian elements. In this paper, we define two types of outer generalized inverses, called pseudo \(e\)-core inverses and pseudo \(f\)-dual core inverses. An element \(a\in R\) is pseudo \(e\)-core invertible if there exist an element \(x\in R\) and some positive integer \(n\) such that \(xax=x,xR=a^nR\) and \(Rx=r(a^n)^\ast e\). Dually, \(a\) is pseudo \(f\)-dual core invertible if there exist an element \(x\in R\) and some positive integer \(m\) such that \(xax =x, Rx=Ra^m\) and \(fxR=(a^m)^\ast R\). Moreover, we investigate both of them for their characterizations and properties. Also, the relations between the pseudo \(e\)-core inverse (resp. the pseudo \(f\)-dual core inverse) and the inverse along an element are given.An SDP method for copositivity of partially symmetric tensors.https://www.zbmath.org/1459.650582021-05-28T16:06:00+00:00"Wang, Chunyan"https://www.zbmath.org/authors/?q=ai:wang.chunyan"Chen, Haibin"https://www.zbmath.org/authors/?q=ai:chen.haibin"Che, Haitao"https://www.zbmath.org/authors/?q=ai:che.haitaoSummary: In this paper, we consider the problem of detecting the copositivity of partially symmetric rectangular tensors. We first propose a semidefinite relaxation algorithm for detecting the copositivity of partially symmetric rectangular tensors. Then, the convergence of the proposed algorithm is given, and it shows that we can always catch the copositivity of given partially symmetric tensors. Several preliminary numerical results confirm our theoretical findings.Properties of the nonnegative solution set of multi-linear equations.https://www.zbmath.org/1459.150052021-05-28T16:06:00+00:00"Xu, Yang"https://www.zbmath.org/authors/?q=ai:xu.yang|xu.yang.3|xu.yang.2|xu.yang.1"Gu, Weizhe"https://www.zbmath.org/authors/?q=ai:gu.weizhe"Huang, Zheng-Hai"https://www.zbmath.org/authors/?q=ai:huang.zheng-haiSummary: Multi-linear equations and tensor complementarity problems are two hot topics in recent years. It is known that the nonnegative solution set of multi-linear equations is a subset of the solution set of the corresponding tensor complementarity problem. In this paper, we first investigate the existence and uniqueness of nonnegative (positive) solution to multi-linear equations induced by some triangular tensors; and then, we discuss the non-existence of nonnegative solution to multi-linear equations induces by \(B\) (\(B_0\)) tensors or strictly diagonally dominant tensors. In addition, we also investigate the boundedness of the nonnegative solution set of multi-linear equations with some structured tensors. The obtained properties of the nonnegative solution set of multi-linear equations give some characteristics on the solution set of tensor complementarity problems.Optimal absorption of acoustic waves by a boundary.https://www.zbmath.org/1459.351212021-05-28T16:06:00+00:00"Magoulès, Frédéric"https://www.zbmath.org/authors/?q=ai:magoules.frederic"Kieu Nguyen, Thi Phuong"https://www.zbmath.org/authors/?q=ai:nguyen.thi-phuong-kieu"Omnes, Pascal"https://www.zbmath.org/authors/?q=ai:omnes.pascal"Rozanova-Pierrat, Anna"https://www.zbmath.org/authors/?q=ai:rozanova-pierrat.annaOn a question of Haemers regarding vectors in the nullspace of Seidel matrices.https://www.zbmath.org/1459.051542021-05-28T16:06:00+00:00"Akbari, S."https://www.zbmath.org/authors/?q=ai:akbari.saeeid|akbari.samira|akbari.saieed|akbari.samin|akbari.shahabeddin|akbari.soheil"Cioabă, S. M."https://www.zbmath.org/authors/?q=ai:cioaba.sebastian-m"Goudarzi, S."https://www.zbmath.org/authors/?q=ai:goudarzi.sobhan"Niaparast, Aidan"https://www.zbmath.org/authors/?q=ai:niaparast.aidan"Tajdini, Artin"https://www.zbmath.org/authors/?q=ai:tajdini.artinSummary: \textit{W. H. Haemers} [MATCH Commun. Math. Comput. Chem. 68, No. 3, 653--659 (2012; Zbl 1289.05290)] asked the following question: If \(S\) is the Seidel matrix of a graph of order \(n\) and \(S\) is singular, does there exist an eigenvector of \(S\) corresponding to 0 which has only \(\pm 1\) elements? In this paper, we construct infinite families of graphs which give a negative answer to this question. One of our constructions implies that for every natural number \(N\), there exists a graph whose Seidel matrix \(S\) is singular such that for any integer vector in the nullspace of \(S\), the absolute value of any entry in this vector is more than \(N\). We also derive some characteristics of vectors in the nullspace of Seidel matrices, which lead to some necessary conditions for the singularity of Seidel matrices. Finally, we obtain some properties of the graphs which affirm the above question.Structured rectangular tensors and rectangular tensor complementarity problems.https://www.zbmath.org/1459.150262021-05-28T16:06:00+00:00"Zeng, Qingyu"https://www.zbmath.org/authors/?q=ai:zeng.qingyu"He, Jun"https://www.zbmath.org/authors/?q=ai:he.jun"Liu, Yanmin"https://www.zbmath.org/authors/?q=ai:liu.yanminSummary: In this paper, some properties of structured rectangular tensors are presented, and the relationship among these structured rectangular tensors is also given. It is shown that all the V-singular values of rectangular P-tensors are positive. Some necessary and/or sufficient conditions for a rectangular tensor to be a rectangular P-tensor are also obtained. A new subclass of rectangular tensors, which is called rectangular S-tensors, is introduced and it is proved that rectangular S-tensors can be defined by the feasible vectors of the corresponding rectangular tensor complementarity problem.The first two maximum ABC spectral radii of bicyclic graphs.https://www.zbmath.org/1459.051962021-05-28T16:06:00+00:00"Yuan, Yan"https://www.zbmath.org/authors/?q=ai:yuan.yan"Du, Zhibin"https://www.zbmath.org/authors/?q=ai:du.zhibinSummary: The ABC matrix of a graph \(G\), proposed by \textit{E. Estrada} [J. Math. Chem. 55, No. 4, 1021--1033 (2017; Zbl 1380.92097)], can be regarded as a weighed version of adjacency matrices of graphs, in which the \((u, v)\)-entry is equal to \(\sqrt{\frac{du+d_v-2}{d_ud_v}}\) if \(uv\) is an edge of a graph \(G\), and 0 otherwise, where \(d_u\) represents the degree of \(u\) in \(G\). The research about ABC spectral radius (largest eigenvalue of ABC matrix) of graphs is rather active in recent years. In this paper, we characterize the bicyclic graphs with the first two maximum ABC spectral radii, which are just the unique two bicyclic graphs of order \(n\) with maximum degree \(n-1\) if \(n\geq 7\).Norm-controlled inversion of Banach algebras of infinite matrices.https://www.zbmath.org/1459.460472021-05-28T16:06:00+00:00"Fang, Qiquan"https://www.zbmath.org/authors/?q=ai:fang.qiquan"Shin, Chang Eon"https://www.zbmath.org/authors/?q=ai:shin.chang-eonThe authors study the properties of the Baskakov-Gohberg-Sjöstrand algebras, which are certain Banach algebras of infinite matrices.
They prove some technical results about the inequalities describing the norm of the inverse of an element of this algebra.
Reviewer: Mart Abel (Tartu)Verified error bounds for real eigenvalues of real symmetric and persymmetric matrices.https://www.zbmath.org/1459.650472021-05-28T16:06:00+00:00"Li, Zhe"https://www.zbmath.org/authors/?q=ai:li.zhe"Wang, Xueqing"https://www.zbmath.org/authors/?q=ai:wang.xueqingSummary: This paper mainly investigates the verification of real eigenvalues of the real symmetric and persymmetric matrices. For a real symmetric or persymmetric matrix, we use \textbf{eig} code in Matlab to obtain its real eigenvalues on the basis of numerical computation and provide an algorithm to compute verified error bound such that there exists a perturbation matrix of the same type within the computed error bound whose exact real eigenvalues are the computed real eigenvalues.The last two days in elementary linear algebra.https://www.zbmath.org/1459.970082021-05-28T16:06:00+00:00"Muench, Donald L."https://www.zbmath.org/authors/?q=ai:muench.donald-l(no abstract)Sedentary quantum walks.https://www.zbmath.org/1459.051712021-05-28T16:06:00+00:00"Godsil, Chris"https://www.zbmath.org/authors/?q=ai:godsil.christopher-davidSummary: Let \(X\) be a graph with adjacency matrix \(A\). The continuous quantum walk on \(X\) is determined by the unitary matrices \(U(t)=\exp(itA)\) (for \(t\in\mathbb{R})\). If \(X\) is the complete graph \(K_n\) and \(a\in V(X)\), then
\[
1-|U(t)_{a,a}|\leq 2/n.
\]
Roughly speaking, this means that a quantum walk on a complete graph stays home with high probability. We say that a family of graphs is sedentary if there is a constant \(c\) such that \(1-|U(t)_{a,a}|\leq c/|V(X)|\) for all \(t\). In this paper we investigate this condition, and produce further examples of sedentary graphs.
A cone over a graph \(X\) is the graph we get by adjoining a new vertex and making it adjacent to each vertex of \(X\). We prove that if \(X\) is the cone over an \(\ell\)-regular graph on \(n\) vertices, then \(|U(t)_{a,a}|\leq\ell^2/(\ell^2+4n)\). It follows that if we choose \(\ell\) and \(n\) such that \(n/\ell^2\to 0\), then a continuous quantum walk starting on the ``conical'' vertex will remain there with probability close to 1. On the other hand, if \(\ell\leq 2\), we show there is a time \(t\) such that all entries in the \(a\)-column of \(U(t)e_a\) have absolute value \(1/\sqrt{n}\). We show that there are large classes of strongly regular graphs such that \(1-|U(t)_{a,a}|\leq c/V(X)\) for some constant \(c\). On the other hand, for Paley graphs on \(n\) vertices, we prove that if \(t=\pi/\sqrt{n}\), then \(|U(t)_{a,a}|\leq 1/n\).Partitioning networks into clusters and residuals with average association.https://www.zbmath.org/1459.820612021-05-28T16:06:00+00:00"Vejmelka, Martin"https://www.zbmath.org/authors/?q=ai:vejmelka.martin"Paluš, Milan"https://www.zbmath.org/authors/?q=ai:palus.milanSummary: We investigate the problem of detecting clusters exhibiting higher-than-average internal connectivity in networks of interacting systems. We show how the average association objective formulated in the context of spectral graph clustering leads naturally to a clustering strategy where each system is assigned to at most one cluster. A residual set is formed of the systems that are not members of any cluster. Maximization of the average association objective leads to a discrete optimization problem, which is difficult to solve, but a relaxed version can be solved using an eigendecomposition of the connectivity matrix. A simple approach to extracting clusters from a relaxed solution is described and developed by applying a variance maximizing solution to the relaxed solution, which leads to a method with increased accuracy and sensitivity. Numerical studies of theoretical connectivity models and of synchronization clusters in a lattice of coupled Lorenz oscillators are conducted to show the efficiency of the proposed approach. The method is applied to an experimentally obtained human resting state functional magnetic resonance imaging dataset and the results are discussed.{
\copyright 2010 American Institute of Physics}Integral unicyclic graphs.https://www.zbmath.org/1459.051622021-05-28T16:06:00+00:00"Braga, Rodrigo O."https://www.zbmath.org/authors/?q=ai:braga.rodrigo-o"Del-Vecchio, Renata R."https://www.zbmath.org/authors/?q=ai:del-vecchio.renata-raposo"Rodrigues, Virgínia M."https://www.zbmath.org/authors/?q=ai:rodrigues.virginia-mSummary: A graph is integral if the spectrum of its adjacency matrix consists entirely of integers. The question about which unicyclic graphs are integral remains open. We contribute to this problem by presenting three infinite families of integral unicyclic graphs. These families are generated by distinct particular solutions of a Diophantine equation. We also show that two integral unicyclic graphs found through a computer search that do not belong to the families we present are unique with their shapes. Necessary conditions for certain unicyclic graphs to be integral are also given.The main eigenvalues of signed graphs.https://www.zbmath.org/1459.051552021-05-28T16:06:00+00:00"Akbari, S."https://www.zbmath.org/authors/?q=ai:akbari.saieed"França, Franscisca A. M."https://www.zbmath.org/authors/?q=ai:franca.franscisca-a-m"Ghasemian, E."https://www.zbmath.org/authors/?q=ai:ghasemian.ebrahim"Javarsineh, M."https://www.zbmath.org/authors/?q=ai:javarsineh.mehrnoosh"de Lima, Leonardo S."https://www.zbmath.org/authors/?q=ai:de-lima.leonardo-sSummary: A signed graph \(G^\sigma\) is an ordered pair \((V(G), E(G))\), where \(V(G)\) and \(E(G)\) are the set of vertices and edges of \(G\), respectively, along with a map \(\sigma\) that signs every edge of \(G\) with +1 or \(-1\). An eigenvalue of the associated adjacency matrix of \(G^\sigma\), denoted by \(A(G^\sigma)\), is a main eigenvalue if the corresponding eigenspace has a non-orthogonal eigenvector to the all-one vector \(j\). We conjectured that for every graph \(G\neq K_2\), \(K_4\backslash\{e\}\), there is a switching \(\sigma\) such that all eigenvalues of \(G^\sigma\) are main. We show that this conjecture holds for every Cayley graphs, distance-regular graphs, vertex and edge-transitive graphs as well as double stars and paths.The proportion of comets in the card game SET.https://www.zbmath.org/1459.000082021-05-28T16:06:00+00:00"May, Dan"https://www.zbmath.org/authors/?q=ai:may.dan"Swenson, Dan"https://www.zbmath.org/authors/?q=ai:swenson.danSummary: We use recurrence relations to answer a question that was posed in this \textit{Journal}: in the game of SET, what is the probability that nine randomly chosen cards form what is called a ``comet''? We then generalize our results to count subsets of an arbitrary finite vector space which sum to the zero vector.Null decomposition of bipartite graphs without cycles of length 0 modulo 4.https://www.zbmath.org/1459.052622021-05-28T16:06:00+00:00"Jaume, Daniel A."https://www.zbmath.org/authors/?q=ai:jaume.daniel-a"Molina, Gonzalo"https://www.zbmath.org/authors/?q=ai:molina.gonzalo"Pastine, Adrián"https://www.zbmath.org/authors/?q=ai:pastine.adrianSummary: In this work we study the null space of bipartite graphs without cycles of length 0 modulo 4 (denoted as \(C_{4k}\)-free bipartite graphs), and its relation to structural properties. We extend the Null Decomposition of trees, introduced by \textit{D. A. Jaume} and \textit{G. Molina} [Discrete Math. 341, No. 3, 836--850 (2018; Zbl 1378.05026)], to \(C_{4k}\)-free bipartite graphs. This decomposition uses the null space of the adjacency matrix of a graph \(G\) to decompose it into two different types of graphs: \(C_N(G)\) and \(C_S(G)\). \(C_N\) has perfect matching number. \(C_S(G)\) has a unique maximum independent set. We obtain formulas for the independence number and the matching number of a \(C_{4k}\)-free bipartite graph using this decomposition. We also show how the number of maximum matchings and the number of maximum independent sets in a \(C_{4k}\)-free bipartite graph are related to its null decomposition.Symmetric completions of cycles and bipartite graphs.https://www.zbmath.org/1459.051642021-05-28T16:06:00+00:00"Cohen, Nir"https://www.zbmath.org/authors/?q=ai:cohen.nir"Pereira, Edgar"https://www.zbmath.org/authors/?q=ai:pereira.edgarSummary: The analysis of symmetric completions of partial matrices associated with a simple graph \(G\), in terms of inertias and minimal rank, simplifies dramatically when \(G\) is bipartite. Essentially, it is equivalent to the analysis of an associated non-symmetric completion problem. All the inertias down to the minimum rank can be obtained, but the minimal rank itself remains NP-hard for general graphs in this class. The class of bipartite graphs includes even cycles but excludes odd cycles. By the above reduction we provide relatively sharp minimal rank estimates for even cycles and discuss some counter-examples raised by odd cycles.Norm and trace estimation with random rank-one vectors.https://www.zbmath.org/1459.650532021-05-28T16:06:00+00:00"Bujanovic, Zvonimir"https://www.zbmath.org/authors/?q=ai:bujanovic.zvonimir"Kressner, Daniel"https://www.zbmath.org/authors/?q=ai:kressner.danielOn the spectral radius of block graphs with prescribed independence number \(\alpha\).https://www.zbmath.org/1459.051662021-05-28T16:06:00+00:00"Conde, Cristian M."https://www.zbmath.org/authors/?q=ai:conde.cristian-m"Dratman, Ezequiel"https://www.zbmath.org/authors/?q=ai:dratman.ezequiel"Grippo, Luciano N."https://www.zbmath.org/authors/?q=ai:grippo.luciano-norbertoSummary: Let \(\mathcal{G}(n,\alpha)\) be the class of block graphs on \(n\) vertices and prescribed independence number \(\alpha\). In this article we prove that the maximum spectral radius \(\rho(G)\), among all graphs \(G\in\mathcal{G}(n,\alpha)\), is reached at a unique graph. As a byproduct we obtain an upper for \(\rho(G)\), when \(G\in\mathcal{G}(n,\alpha)\).On the spectrum of hypergraphs.https://www.zbmath.org/1459.051582021-05-28T16:06:00+00:00"Banerjee, Anirban"https://www.zbmath.org/authors/?q=ai:banerjee.anirbanSummary: Here, we introduce different connectivity matrices and study their eigenvalues to explore various structural properties of a general hypergraph. We investigate how the diameter, connectivity and vertex chromatic number of a hypergraph are related to the spectrum of these matrices. Different properties of a regular hypergraph are also characterized by the same. Cheeger constant on a hypergraph is defined and its spectral bounds have been derived for a connected general hypergraph. Random walk on a general hypergraph can also be well studied by analyzing the spectrum of the transition probability operator defined on the hypergraph. We also introduce Ricci curvature on a general hypergraph and study its relation with the hypergraph spectra.Componentwise perturbation analysis of the Schur decomposition of a matrix.https://www.zbmath.org/1459.650512021-05-28T16:06:00+00:00"Petkov, Petko H."https://www.zbmath.org/authors/?q=ai:petkov.petko-hrOn cardinality of complementarity spectra of connected graphs.https://www.zbmath.org/1459.051862021-05-28T16:06:00+00:00"Seeger, Alberto"https://www.zbmath.org/authors/?q=ai:seeger.alberto"Sossa, David"https://www.zbmath.org/authors/?q=ai:sossa.davidSummary: This work deals with complementarity spectra of connected graphs and, specifically, with the associated concept of spectral capacity of a finite set of connected graphs. The cardinality of the complementarity spectrum of a connected graph \(G\) serves as lower bound for the number of connected induced subgraphs of \(G\). Motivated by this observation, we establish various results on cardinality of complementarity spectra. Special attention is paid to the asymptotic behavior of spectral capacities as the number of vertices goes to infinity.\(\lambda\)-core distance partitions.https://www.zbmath.org/1459.051812021-05-28T16:06:00+00:00"Mifsud, Xandru"https://www.zbmath.org/authors/?q=ai:mifsud.xandruSummary: The \(\lambda\)-core vertices of a graph correspond to the non-zero entries of some eigenvector of \(\lambda\) for a universal adjacency matrix \(\mathfrak{U}\) of the graph. We define a partition of the vertex set \(V\) based on the \(\lambda\)-core vertex set and its neighbourhoods at a distance \(r\), and give a number of results relating the structure of the graph to this partition. For such partitions, we also define an entropic measure for the information content of a graph, related to every distinct eigenvalue \(\lambda\) of \(\mathfrak{U}\), and discuss its properties and potential applications.On improved universal estimation of exponents of digraphs.https://www.zbmath.org/1459.051032021-05-28T16:06:00+00:00"Fomichev, V. M."https://www.zbmath.org/authors/?q=ai:fomichev.v-mSummary: An improved formula for universal estimation of exponent is obtained for \(n\)-vertex primitive digraphs. A previous formula by \textit{A. L. Dulmage} and \textit{N. S. Mendelsohn} [Ill. J. Math. 8, 642--656 (1964; Zbl 0125.00706)] is based on a system \(\hat{C}\) of directed circuits \(C_1,\ldots,C_m\), which are held in a graph and have lengths \(l_1,\ldots,l_m\) with \(\gcd(l_1,\ldots,l_m)=1\). A new formula is based on a similar circuit system \(\hat{C} \), where \(\gcd(l_1,\ldots,l_m)=d\geq 1\). Also, the new formula uses \(r_{i,j}^{s/d}(\hat{C})\), that is the length of the shortest path from \(i\) to \(j\) going through the circuit system \(\hat{C}\) and having the length which is comparable to \(s\) modulo \(d, s=0,\ldots,d-1\). It is shown, that \(\text{exp}\,\Gamma\leq 1+\hat{F}(L(\hat{C}))+R(\hat{C})\), where \(\hat{F}(L)=d\cdot F(l_1/d,\ldots, l_m/d)\) and \(F(a_1,\ldots,a_m)\) is the Frobenius number, \(R(\hat{C})=\max_{(i,j)}\max_s\{r_{i,j}^{s/d}(\hat{C})\} \). For some class of \(2k\)-vertex primitive digraphs, it is proved, that the improved formula gives the value of estimation \(2k\), and the previous formula gives the value of estimation \(3k-2\).The characterization of generalized Jordan centralizers on triangular algebras.https://www.zbmath.org/1459.160292021-05-28T16:06:00+00:00"Chen, Quanyuan"https://www.zbmath.org/authors/?q=ai:chen.quanyuan"Fang, Xiaochun"https://www.zbmath.org/authors/?q=ai:fang.xiaochun"Li, Changjing"https://www.zbmath.org/authors/?q=ai:li.changjingSummary: In this paper, it is shown that if \(\mathcal{T} = \mathrm{Tri}(\mathcal{A}, \mathcal{M}, \mathcal{B})\) is a triangular algebra and \(\phi\) is an additive operator on \(\mathcal{T}\) such that \((m + n + k + l) \phi(T^2) -(m \phi(T) T + n T \phi(T) + k \phi(I) T^2 + l T^2 \phi(I)) \in \mathbb{F} I\) for any \(T \in \mathcal{T}\), then \(\phi\) is a centralizer. It follows that an \((m, n)\)-Jordan centralizer on a triangular algebra is a centralizer.On upper bound of permanents.https://www.zbmath.org/1459.150092021-05-28T16:06:00+00:00"Efimov, D. B."https://www.zbmath.org/authors/?q=ai:efimov.dmitry-borisovichSummary: By Jurkat-Rayser method we obtain an upper bound of the permanent of an arbitrary real nonnegative matrix of order 3 in general and special case. We give also a comparision with the similar upper bound of the permanent obtained by another methods.Positive definiteness for 4th order symmetric tensors and applications.https://www.zbmath.org/1459.150272021-05-28T16:06:00+00:00"Song, Yisheng"https://www.zbmath.org/authors/?q=ai:song.yisheng.1|song.yishengSummary: In particle physics, the vacuum stability of scalar potentials is to check positive definiteness (or copositivity) of its coupling tensors, and such a coupling tensor is a 4th order and symmetric tensor. In this paper, we mainly discuss precise expressions of positive definiteness of 4th order tensors. More specifically, two analytically sufficient conditions of positive definiteness for 4th order 2 dimensional symmetric tensors are given by reducing orders of tensors, and applying these conclusions, some sufficient conditions for the positive definiteness of 4th order 3 dimensional symmetric tensors are derived. We also present several other sufficient conditions for the positive definiteness of 4th order 3 dimensional symmetric tensors. Finally, we test and verify the vacuum stability of general scalar potentials of two real singlet scalar fields and the Higgs boson by using these results.Vertex distinction with subgraph centrality: a proof of Estrada's conjecture and some generalizations.https://www.zbmath.org/1459.053102021-05-28T16:06:00+00:00"Ballini, Francesco"https://www.zbmath.org/authors/?q=ai:ballini.francesco"Deniskin, Nikita"https://www.zbmath.org/authors/?q=ai:deniskin.nikitaSummary: Centrality measures are used in network science to identify the most important vertices for transmission of information and dynamics on a graph. One of these measures, introduced by \textit{E. Estrada} and \textit{J. A. Rodríguez-Velázquez} [``Subgraph centrality in complex networks'', Phys. Rev. E 71, Article ID 056103, 9 p. (2005; \url{doi:10.1103/PhysRevE.71.056103})], is the \(\beta\)-subgraph centrality, which is based on the exponential of the matrix \(\beta A\), where \(A\) is the adjacency matrix of the graph and \(\beta\) is a real parameter (``inverse temperature''). We prove that for algebraic \(\beta\), two vertices with equal \(\beta\)-subgraph centrality are necessarily cospectral. We further show that two such vertices must have the same degree and eigenvector centralities. Our results settle a conjecture of Estrada and a generalization of it due to \textit{K. Kloster} et al. [Linear Algebra Appl. 546, 115--121 (2018; Zbl 1391.05169)]. We also discuss possible extensions of our results.Asymptotic values of four Laplacian-type energies for matrices with degree-distance-based entries of random graphs.https://www.zbmath.org/1459.051792021-05-28T16:06:00+00:00"Li, Xueliang"https://www.zbmath.org/authors/?q=ai:li.xueliang"Li, Yiyang"https://www.zbmath.org/authors/?q=ai:li.yiyang"Wang, Zhiqian"https://www.zbmath.org/authors/?q=ai:wang.zhiqianThis paper studies asymptotic values of Laplacian-type energies for matrices with degree-distance-based entries of the Erdös-Rényi random graph \(G(n,p)\). Let a real symmetric function \(f\) be given over \(G\), let \(\operatorname{LEL}_f(G)\) be the associated weighted Laplacian-energy like invariant and \(\operatorname{IE}_f(G)\) be the weighted incidence energy. Let \(f_1(d_i,d_j)\) and \(f_2(d_i,d_j)\) be two symmetric functions satisfying \(f_1((1+o(1))np,(1+o(1))np)=(1+o(1))f_1(np,np)\) and \(f_s((1+o(1))np,(1+o(1))np)=(1+o(1))f_2(np,np)\), then it is shown that almost surely
\begin{itemize}
\item[(i)] if \(f_1(np,np)/f_2(np,np)\rightarrow\infty\) or \(f_1(np,np)/f_2(np,np)\rightarrow-\infty\), then \(\operatorname{LEL}_f(G(n,p))=\sqrt{|f_1(np,np)|}(\sqrt{p}+o(1))n^{3/2}\) and \(\operatorname{IE}_f(G(n,p))=\sqrt{|f_1(np,np)|}(\sqrt{p}+o(1))n^{3/2}\);
\item[(ii)] if \(f_1(np,np)/f_2(np,np)\rightarrow C\) for some constant \(C\), then \(\operatorname{LEL}_f(G(n,p))=\sqrt{|f_2(np,np)|}\) \((\sqrt{1+(C-1)p}+o(1))\) \(n^{3/2}\) and \(\operatorname{LEL}_f(G(n,p))=\sqrt{|f_2(np,np)|}\) \((\sqrt{1+(C-1)p}+o(1))\) \(n^{3/2}\).
\end{itemize}
Reviewer: Yilun Shang (Newcastle)Edge-matching graph contractions and their interlacing properties.https://www.zbmath.org/1459.051782021-05-28T16:06:00+00:00"Leiter, Noam"https://www.zbmath.org/authors/?q=ai:leiter.noam"Zelazo, Daniel"https://www.zbmath.org/authors/?q=ai:zelazo.danielSummary: For a given graph \(\mathcal{G}\) of order \(n\) with \(m\) edges, and a real symmetric matrix associated to the graph, \(M(\mathcal{G})\in\mathbb{R}^{n\times n}\), the interlacing graph reduction problem is to find a graph \(\mathcal{G}_r\) of order \(r<n\) such that the eigenvalues of \(M(\mathcal{G}_r)\) interlace the eigenvalues of \(M(\mathcal{G})\). Graph contractions over partitions of the vertices are widely used as a combinatorial graph reduction tool. In this study, we derive a graph reduction interlacing theorem based on subspace mappings and the minmax theory. We then define a class of edge-matching graph contractions and show how two types of edge-matching contractions provide Laplacian and normalized Laplacian interlacing. An \(\mathcal{O}(mn)\) algorithm is provided for finding a normalized Laplacian interlacing contraction and an \(\mathcal{O}(n^2+nm)\) algorithm is provided for finding a Laplacian interlacing contraction.Factorizations for a class of multivariate polynomial matrices.https://www.zbmath.org/1459.150142021-05-28T16:06:00+00:00"Lu, Dong"https://www.zbmath.org/authors/?q=ai:lu.dong"Wang, Dingkang"https://www.zbmath.org/authors/?q=ai:wang.dingkang"Xiao, Fanghui"https://www.zbmath.org/authors/?q=ai:xiao.fanghuiSummary: This paper investigates how to factorize a class of multivariate polynomial matrices. We prove that an \(l\times m\) multivariate polynomial matrix admits a matrix factorization with respect to a given polynomial if the polynomial and all the \((l-1)\times (l-1)\) reduced minors of the matrix generate a unit ideal. This result is a generalization of a theorem in [\textit{J. Liu} et al., Circuits Syst. Signal Process. 30, No. 3, 553--566 (2011; Zbl 1213.93034)]. Based on three main theorems presented in the paper and a constructive algorithm proposed by \textit{Z. Lin} et al. [Circuits Syst. Signal Process. 20, No. 6, 601--618 (2001; Zbl 1024.93014)], we give an algorithm which can be used to factorize more multivariate polynomial matrices. In addition, an illustrative example is given to show the effectiveness of the proposed algorithm.A Coxeter spectral classification of positive edge-bipartite graphs. II: Dynkin type \(\mathbb{D}_n\).https://www.zbmath.org/1459.051132021-05-28T16:06:00+00:00"Simson, Daniel"https://www.zbmath.org/authors/?q=ai:simson.danielSummary: We continue the Coxeter spectral study of finite positive edge-bipartite signed (multi)graphs \(\Delta\) (bigraphs, for short), with \(n\geq 2\) vertices started in [the author, SIAM J. Discrete Math. 27, No. 2, 827--854 (2013; Zbl 1272.05072)] and developed in [the author, Linear Algebra Appl. 557, 105--133 (2018; Zbl 1396.05049)]. We do it by means of the non-symmetric Gram matrix \(\check{G}_{\Delta}\in \mathbb{M}_n(\mathbb{Z})\) defining \(\Delta\), its Gram quadratic form \(q_{\Delta}:\mathbb{Z}^n\to\mathbb{Z}\), \(v\mapsto v\cdot\check{G}_{\Delta}\cdot v^{tr}\) (that is positive definite, by definition), the complex spectrum \(\operatorname{specc}_{\Delta}\subset\mathcal{S}^1:=\{z\in\mathbb{C},|z|=1\}\) of the Coxeter matrix \(\operatorname{Cox}_{\Delta}:=-\check{G}_{\Delta}\cdot \check{G}_{\Delta}^{-tr}\in\mathbb{M}_n(\mathbb{Z})\), called the Coxeter spectrum of \(\Delta\), and the Coxeter polynomial \(\operatorname{cox}_{\Delta}(t):=\det(t\cdot E-\operatorname{Cox}_{\Delta})\in\mathbb{Z} [t]\). One of the aims of the Coxeter spectral analysis is to classify the connected bigraphs \(\Delta\) with \(n\geq 2\) vertices up to the \(\ell\)-weak Gram \(\mathbb{Z}\)-congruence \(\Delta\sim_{\ell\mathbb{Z}} \Delta^\prime\) and up to the strong Gram \(\mathbb{Z}\)-congruence \(\Delta\approx_{\mathbb{Z}}\Delta^\prime\), where \(\Delta\sim_{\ell\mathbb{Z}}\Delta^\prime\) (resp. \(\Delta\approx_{\mathbb{Z}}\Delta^\prime)\) means that \(\det \check{G}_{\Delta}=\det \check{G}_{\Delta^\prime}\) and \(G_{\Delta^\prime}=B^{tr}\cdot G_{\Delta}\cdot B\) (resp. \(\check{G}_{\Delta^\prime}=B^{tr}\cdot \check{G}_{\Delta}\cdot B)\), for some \(B\in\mathbb{M}_n(\mathbb{Z})\) with \(\det B=\pm 1\), where \(G_{\Delta}:=\frac{1}{2}[\check{G}_{\Delta}+\check{G}_{\Delta}^{tr}]\in \mathbb{M}_n(\frac{1}{2}\mathbb{Z})\).
Here we study connected signed simple graphs \(\Delta\), with \(n\geq 2\) vertices, that are positive, i.e., the symmetric Gram matrix \(G_{\Delta}\in\mathbb{M}_n(\frac{1}{2}\mathbb{Z})\) of \(\Delta\) is positive definite. It is known that every such a signed graph is \(\ell\)-weak Gram \(\mathbb{Z}\)-congruent with a unique simply laced Dynkin graph \(\operatorname{Dyn}_{\Delta} \in\{\mathbb{A}_n,\mathbb{D}_n,n\geq 4,\mathbb{E}_6, \mathbb{E}_7,\mathbb{E}_8\}\), called the Dynkin type of \(\Delta\). A classification up to the strong Gram \(\mathbb{Z}\)-congruence \(\Delta\approx_{\mathbb{Z}}\Delta^\prime\) is still an open problem and only partial results are known. In this paper, we obtain such a classification for the positive signed simple graphs \(\Delta\) of Dynkin type \(\mathbb{D}_n\) by means of the family of the signed graphs \(\mathcal{D}_n^{(1)}=\mathbb{D}_n, \mathcal{D}_n^{(2)},\dots,\mathcal{D}_n^{(r_n)}\) constructed in Section 2, for any \(n\geq 4\), where \(r_n=\llcorner n/2\lrcorner\). More precisely, we prove that any connected signed simple graph of Dynkin type \(\mathbb{D}_n\), with \(n\geq 4\) vertices, is strongly \(\mathbb{Z}\)-congruent with a signed graph \(\mathcal{D}_n^{(s)}\), for some \(s\leq r_n\), and its Coxeter polynomial \(\operatorname{cox}_{\Delta}(t)\) is of the form \((t^s+1)(t^{n-s}+1)\). We do it by a matrix morsification type reduction to the classification of the conjugacy classes in the integral orthogonal group \(\operatorname{O}(n, \mathbb{Z})\) of the integer orthogonal matrices \(C\), with \(\det(E-C)=4\).Pairs of commuting nilpotent operators with one-dimensional intersection of kernels and matrices commuting with a Weyr matrix.https://www.zbmath.org/1459.150132021-05-28T16:06:00+00:00"Bondarenko, Vitalij M."https://www.zbmath.org/authors/?q=ai:bondarenko.vitalij-m"Futorny, Vyacheslav"https://www.zbmath.org/authors/?q=ai:futorny.vyacheslav-m"Petravchuk, Anatolii P."https://www.zbmath.org/authors/?q=ai:petravchuk.anatolii-p"Sergeichuk, Vladimir V."https://www.zbmath.org/authors/?q=ai:sergeichuk.vladimir-vasilevich.1The authors deal with the problem of classifying pairs \((A, B)\) of commuting nilpotent operators on a vector space. Such a problem has its origin in the paper of \textit{I. M. Gel'fand} and \textit{V. A. Ponomarev} [Funkts. Anal. Prilozh. 3, No. 4, 81--82 (1969; Zbl 0204.45301)] where they proved that classifying pairs \((M, N)\) of commuting nilpotent matrices under similarity transformations includes the problem of classifying \(t\)-tuples of matrices with any \(t\) under similarity transformations
\[
(A_1, \ldots, A_t) \rightarrow (S^{-1} A_1 S, \ldots, S^{-1} A_t S), \qquad S {\mbox{ is nonsingular}}.
\]
Using Belitskii's algorithm, the authors reduce \((M, N)\) by similarity transformations to some simple pair \((W_M, B)\), where \(W_M\) is the Weyr canonical form of \(M\). They also show that uniqueness of the pair \((W_M, B)\) can be proved only if the Jordan canonical form of \(M\) is a direct sum of Jordan blocks of the same size and the field \(\mathbb{F}\) is of zero characteristic. Finally in order to describe the structure of the matrix \(B\), the authors describe the form of all matrices that commute with a Weyr matrix.
Reviewer: Ninoslav Truhar (Osijek)Classes of nonbipartite graphs with reciprocal eigenvalue property.https://www.zbmath.org/1459.051592021-05-28T16:06:00+00:00"Barik, Sasmita"https://www.zbmath.org/authors/?q=ai:barik.sasmita"Pati, Sukanta"https://www.zbmath.org/authors/?q=ai:pati.sukantaSummary: Let \(G\) be a simple connected graph and \(A(G)\) be the adjacency matrix of \(G\). The graph \(G\) is said to have the reciprocal eigenvalue property (R) if \(A(G)\) is nonsingular and \(\frac{1}{\lambda}\) is an eigenvalue of \(A(G)\) whenever \(\lambda\) is an eigenvalue of \(A(G)\). Further, if \(\lambda\) and \(\frac{1}{\lambda}\) have the same multiplicity, for each eigenvalue \(\lambda\), then \(G\) is said to have the strong reciprocal eigenvalue property (SR). Till date, all the classes of bipartite graphs that are found to have property (R), are found to have property (SR) and it is not known whether these two properties are equivalent even for the bipartite graphs with a unique perfect matching. Among nonbipartite graphs, there is only one known graph class for which these two properties are not equivalent. In this article, we construct some more classes of nonbipartite graphs with property (R) but not (SR).On the Smith normal form of walk matrices.https://www.zbmath.org/1459.051912021-05-28T16:06:00+00:00"Wang, Wei"https://www.zbmath.org/authors/?q=ai:wang.wei.17|wang.wei.25|wang.wei.8|wang.wei.21|wang.wei.26|wang.wei.18|wang.wei.29|wang.wei.15|wang.wei.24|wang.wei.28|wang.wei.30|wang.wei.12|wang.wei.19|wang.wei.3|wang.wei.13|wang.wei.16|wang.wei.5|wang.wei.27|wang.wei.1|wang.wei.2|wang.wei.9|wang.wei.20|wang.wei.23Summary: Let \(G\) be a graph with \(n\) vertices. The walk matrix \(W(G)\) of \(G\) is the matrix \([e,Ae,\dots,A^{n-1}e]\), where \(A\) is the adjacency matrix of \(G\) and \(e\) is the all-one vector. Let \(W\) be a walk matrix of order \(n\). We show that at most \(\lfloor \frac{n}{2}\rfloor\) invariant factors of \(W\) are congruent to 2 modulo 4. As a consequence, it is proved that, for any \(n \times n\) walk matrix \(W\) with 2-rank \(r\), the determinant of \(W\) is always a multiple of \(2^{\lceil\frac{3n-4r}{2}\rceil}\). Moreover, if \(2^{-\lceil\frac{3n-4r}{2}\rceil}\det W\) is odd and square-free, then the Smith normal form of \(W\) can be recovered uniquely from the triple \((n,r,\det W)\).The \(\alpha\)-normal labeling for generalized directed uniform hypergraphs.https://www.zbmath.org/1459.052902021-05-28T16:06:00+00:00"Li, Wei"https://www.zbmath.org/authors/?q=ai:li.wei.11|li.wei.2|li.wei.10|li.wei.9|li.wei.3|li.wei.7|li.wei.4|li.wei|li.wei.8|li.wei-wayne"Liu, Lele"https://www.zbmath.org/authors/?q=ai:liu.leleSummary: Let \(\pi=(\nu_1,\dots,\nu_d)\) be an ordered partition of the integer \(m\geq 1\), where \(\sum_{i=1}^d\nu_i=m\) and \(\nu_i\in\mathbb{Z}^+\) for all \(i\in [d]\). A \(\pi\)-directed \(m\)-uniform hypergraph \(G\) consists of a finite vertex set \(V(G)\) and a collection of edges \(E(G)\). Each edge is an ordered tuple, \(e=(S_1(e),S_2(e),\dots,S_d(e))\), of disjoint subsets of vertices such that \(|S_i|=\nu_i\), for all \(i\in [d]\). For each edge \(e,|\bigcup_{i=1}^dS_i(e)|=m\). Moreover, \(\bigcup_{e\in E(G)} S_i(e)=S_i(G)\) and \(|S_i(G)|=n_i\). Given \(\mathfrak{p}=(p_1,p_2,\dots,p_d)\in(1,\infty)^d\), the \(\mathfrak{p}\)-spectral radius of \(G\) is defined as
\[
\lambda_{\mathfrak{p}}(G):=\max\limits_{\|\mathfrak{x}_i\|_{p_i}=1,i\in [d]}\sum\limits_{e\in E(G)}\prod\limits_{i=1}^d\prod\limits_{v\in S_i(e)}x_{i,v},
\]
where \(\mathfrak{x}_i=(x_{i,1},x_{i,2},\dots,x_{i,n_i})\in \mathbb{R}^{n_i}\). In this paper, we develop the \(\alpha\)-normal labeling method for calculating \(\lambda_{\mathfrak{p}}(G)\) and some related properties are given. Moreover, we give a new lower bound of the \(\mathfrak{p}\)-spectral radius of \(\pi\)-directed \(m\)-uniform hypergraphs for the case \(\sum_{i=1}^d \frac{\nu_i}{p_i}>1\) by using the inverse Hölder's inequality.Equivalence between GLT sequences and measurable functions.https://www.zbmath.org/1459.470092021-05-28T16:06:00+00:00"Barbarino, Giovanni"https://www.zbmath.org/authors/?q=ai:barbarino.giovanniAuthor's abstract: The theory of generalized locally Toeplitz (GLT) sequences of matrices has been developed in order to study the asymptotic behaviour of particular spectral distributions when the dimension of the matrices tends to infinity. Key concepts in this theory are the notions of approximating classes of sequences (a.c.s.)\ and spectral symbols that lead to defining a metric structure on the space of matrix sequences and provide a link with the measurable functions. In this paper, we prove additional results regarding theoretical aspects, such as the completeness of the matrix sequences space with respect to the metric a.c.s.\ and the identification of the space of GLT sequences with the space of measurable functions.
Reviewer: Tomasz Natkaniec (Gdańsk)Proof of a conjecture on distance energy change of complete multipartite graph due to edge deletion.https://www.zbmath.org/1459.051882021-05-28T16:06:00+00:00"Sun, Shaowei"https://www.zbmath.org/authors/?q=ai:sun.shaowei"Das, Kinkar Chandra"https://www.zbmath.org/authors/?q=ai:das.kinkar-chandraSummary: Let \(K_{n_1,n_2,\dots,n_k}\) be a complete \(k\)-partite graph with \(k\geq 2\) and \(n_i\geq 2\) for \(i=1,2,\dots,k\). The Turán graph \(T(n, k)\) is a complete \(k\)-partite graph of \(n\) vertices with sizes of partitions as equal as possible. The distance energy \(E_D(G)\) of a graph \(G\) is defined as the sum of absolute values of distance eigenvalues of the graph \(G\). \textit{A. Varghese} et al. [ibid. 553, 211--222 (2018; Zbl 1391.05179)] conjectured that
\[
E_D(K_{n_1,n_2,\dots,n_k})<E_D(K_{n_1,n_2,\dots,n_k}-e),
\]
where \(e\) is any edge of \(K_{n_1,n_2,\dots,n_k}\) and proved that the above relation holds for \(k=2\). Very recently, \textit{G.-X. Tian} et al. [ibid. 584, 438--457 (2020; Zbl 1426.05111)] confirmed that the above conjecture holds for \(T(n,k)\) with \(n\equiv 0\pmod{k}\) and \(T(n,3)\). They also mentioned a weaker conjecture as follows:
\[
E_D(T(n, k))<E_D(T(n, k)-e),
\]
where \(e\) is any edge of \(T(n, k)\) and \(k\geq 2\), \(n\geq 2k\). In this paper, we confirm that the former conjecture is true for \(k\geq 3\) and then the latter conjecture follows immediately.On convergence rate of the randomized Gauss-Seidel method.https://www.zbmath.org/1459.650402021-05-28T16:06:00+00:00"Bai, Zhong-Zhi"https://www.zbmath.org/authors/?q=ai:bai.zhongzhi"Wang, Lu"https://www.zbmath.org/authors/?q=ai:wang.lu|wang.lu.1|wang.lu.3|wang.lu.4|wang.lu.2"Wu, Wen-Ting"https://www.zbmath.org/authors/?q=ai:wu.wentingSummary: The Gauss-Seidel and Kaczmarz methods are two classic iteration methods for solving systems of linear equations, which operate in column and row spaces, respectively. Utilizing the connections between these two methods and imitating the exact analysis of the mean-squared error for the randomized Kaczmarz method, we conduct an exact closed-form formula for the mean-squared residual of the iterate generated by the randomized Gauss-Seidel method. Based on this new formula, we further estimate an upper bound for the convergence rate of the randomized Gauss-Seidel method. Theoretical analysis and numerical experiments show that this bound measurably improves the existing ones. Moreover, these theoretical results are also extended to the more general extrapolated randomized Gauss-Seidel method.A reduction formula for the characteristic polynomial of hypergraph with pendant edges.https://www.zbmath.org/1459.052222021-05-28T16:06:00+00:00"Chen, Lixiang"https://www.zbmath.org/authors/?q=ai:chen.lixiang"Bu, Changjiang"https://www.zbmath.org/authors/?q=ai:bu.changjiangSummary: In this paper, we give a reduction formula for the characteristic polynomial of \(k\)-uniform hypergraphs with pendant edges, and use the reduction formula to derive the explicit expression for the characteristic polynomial and all distinct eigenvalues of \(k\)-uniform loose hyperpaths.On real algebras generated by positive and nonnegative matrices.https://www.zbmath.org/1459.150222021-05-28T16:06:00+00:00"Kolegov, N. A."https://www.zbmath.org/authors/?q=ai:kolegov.n-aLet \(M_{n}(\mathbb{R})\) be the algebra of \(n\times n\) real matrices. A matrix
\(A=[a_{ij}]\in M_{n}(\mathbb{R})\) is called positive (respectively,
nonnegative) if all \(a_{ij}>0\) (respectively, \(a_{ij}\geq0\)). Two matrices \(A\)
and \(B\) are called semi-commuting if one of the additive commutators \([A,B]\)
or \([B,A]\) is nonnegative (see [\textit{R. Drnovšek}, Positivity 22, No. 3, 815--828 (2018; Zbl 1396.15016)]). This paper considers
subalgebras \(\mathcal{A}\) of \(M_{n}(\mathbb{R)}\) which are similar to
algebras that can be generated by a set of positive matrices. The following claims give a flavour of the results.
(Corollary 2.7). If \(\mathcal{A}\) is unital and contains a positive matrix then \(\mathcal{A}\) is
positively generated.
(Theorem 3.5). If \(\mathcal{A}\) is a unital algebra,
then the following are equivalent:
(1) \(\mathcal{A}\) is similar to a
positively generated algebra;
(2) \(\mathcal{A}\) contains a matrix with a
simple real eigenvalue;
(3) \(\mathcal{A}\) contains an idempotent matrix of
rank \(1\);
(4) \(\mathcal{A}\) is similar to an algebra in upper triangular block form
\[
\left[
\begin{array}
[c]{ccc}
\mathcal{B}_{1} & \ast & \ast\\
0 & M_{k}(\mathbb{R)} & \ast\\
0 & 0 & \mathcal{B}_{2}
\end{array}
\right]
\]
where \(k\geq1\) and the \(\mathcal{B}_{i}\) are matrix algebras (one or other may not appear).
(Corollary 4.6 and Corollary 4.8). If \(\mathcal{A}\) contains a
matrix \(A\) with a real eigenvalue, then the centralizer \(C(A)\) is similar to a
nonnegatively generated algebra, and the algebra
\(\left\langle A\right\rangle_{\mathrm{Alg}}\)
can be generated by a matrix which is similar to a
nonnegative matrix.
(Theorem 5.1). Every incidence algebra \(\mathcal{A}
\subseteq M_{n}(\mathbb{R})\) is generated by two nonnegative semi-commuting
matrices.
(Theorem 5.2). For each integer \(k\) with
\(n\leq k\leq\frac{1}{2}n(n+1)\) there is an incidence algebra contained in \(M_{n}(\mathbb{R})\) of dimension
\(k\).
The last two theorems answer a question raised in [\textit{M. Kandić} and \textit{K. Šivic}, Linear Algebra Appl. 512, 136--161 (2017; Zbl 1353.15016)].
Reviewer: John D. Dixon (Ottawa)Continuity of the core-EP inverse and its applications.https://www.zbmath.org/1459.150062021-05-28T16:06:00+00:00"Gao, Yuefeng"https://www.zbmath.org/authors/?q=ai:gao.yuefeng"Chen, Jianlong"https://www.zbmath.org/authors/?q=ai:chen.jianlong"Patrício, Pedro"https://www.zbmath.org/authors/?q=ai:patricio.pedroSummary: In this paper, firstly we study the continuity of the core-EP inverse without explicit error bounds by virtue of two methods. One is the rank equality, followed from the classical generalized inverse. The other one is matrix decomposition. The continuity of the core inverse can be derived as a particular case. Secondly, we study perturbation bounds for the core-EP inverse under prescribed conditions. Perturbation bounds for the core inverse can be derived as a particular case. Also, as corollaries, the sufficient (and necessary) conditions for the continuity of the core-EP inverse are obtained. Thirdly, a numerical example is illustrated to compare derived upper bounds. Finally, an application to semistable matrices is provided.On the solution of the nonsymmetric T-Riccati equation.https://www.zbmath.org/1459.150152021-05-28T16:06:00+00:00"Benner, Peter"https://www.zbmath.org/authors/?q=ai:benner.peter"Palitta, Davide"https://www.zbmath.org/authors/?q=ai:palitta.davideSummary: The nonsymmetric T-Riccati equation is a quadratic matrix equation where the linear part corresponds to the so-called T-Sylvester or T-Lyapunov operator that has previously been studied in the literature. It has applications in macroeconomics and policy dynamics. So far, it presents an unexplored problem in numerical analysis, and both theoretical results and computational methods are lacking in the literature. In this paper we provide some sufficient conditions for the existence and uniqueness of a nonnegative minimal solution, namely the solution with component-wise minimal entries. Moreover, the efficient computation of such a solution is analyzed. Both the small-scale and large-scale settings are addressed, and Newton-Kleinman-like methods are derived. The convergence of these procedures to the minimal solution is proven, and several numerical results illustrate the computational efficiency of the proposed methods.Topological properties of \(J\)-orthogonal matrices. II.https://www.zbmath.org/1459.150302021-05-28T16:06:00+00:00"Motlaghian, Sara M."https://www.zbmath.org/authors/?q=ai:motlaghian.sara-m"Armandnejad, Ali"https://www.zbmath.org/authors/?q=ai:armandnejad.ali"Hall, Frank J."https://www.zbmath.org/authors/?q=ai:hall.frank-jSummary: This paper is a continuation of the authors' article [ibid. 66, No. 12, 2524--2533 (2018; Zbl 1430.15023)]. Let \(\mathbf{M}_n\) be the set of all \(n \times n\) real matrices. A matrix \(J \in \mathbf{M}_n\) is said to be a signature matrix if \(J\) is diagonal and its diagonal entries are \(\pm 1\). If \(J\) is a signature matrix, a nonsingular matrix \(A \in \mathbf{M}_n\) is said to be a \(J\)-orthogonal matrix if \(A^\top JA = J\). Let \(\Omega_n\) be the set of all \(n \times n\), \(J\)-orthogonal matrices. In this paper some further interesting properties of these matrices are obtained. In particular, an open question stated in the preceding article about \(\Omega_n\) is answered. Proposition 3.2 on the characterization of \(J\)-orthogonal matrices in the paper [\textit{N. J. Higham}, SIAM Rev. 45, No. 3, 504--519 (2003; Zbl 1034.65026)] is again heavily used. The standard linear operators \(T : \mathbf{M}_n \rightarrow \mathbf{M}_n)) \) that strongly preserve \(J\)-orthogonal matrices, i.e. \(T(A)\) is \(J\)-orthogonal if and only if \(A\) is \(J\)-orthogonal are characterized.The maximum rank of \(2 \times \cdots \times 2\) tensors over \(\mathbb{F}_2\).https://www.zbmath.org/1459.150252021-05-28T16:06:00+00:00"Stavrou, Stavros Georgios"https://www.zbmath.org/authors/?q=ai:stavrou.stavros-georgios"Low, Richard M."https://www.zbmath.org/authors/?q=ai:low.richard-mSummary: We determine that the maximum rank of an order-\(n\) (\(\geq 2\)) tensor with format \(2 \times \cdots \times 2\) over the finite field \(\mathbb{F}_2\) is \(2 \cdot 3^{n/2-1}\) for even \(n\), and \(3^{[n/2]}\) for odd \(n\). Since tensor rank is non-increasing upon taking field extensions, \(\mathbb{F}_2\) gives the largest rank attainable for this tensor format. We also determine a maximum rank canonical form and compute its orbit under the action of the symmetry group \(\mathrm{GL}_2(\mathbb{F}_2)^{\times n}\), and prove that this is the unique maximum rank canonical form, for even \(n \geq 2\).On the solvability of interval max-min matrix equations.https://www.zbmath.org/1459.150192021-05-28T16:06:00+00:00"Myšková, Helena"https://www.zbmath.org/authors/?q=ai:myskova.helena"Plavka, Ján"https://www.zbmath.org/authors/?q=ai:plavka.janMax-min algebra is a triple \((\mathcal{I}, \oplus, \otimes),\) where \(\mathcal{I}=[O, I]\) is a linearly ordered set with the least element \(O\) and the greatest element \(I\) and \(\oplus, \otimes\) are binary operations defined for two elements with \(\max\) and \(\min\). These binary operations are extended to matrices and vectors as in classical algebra. The authors denote the set of all \(m \times n\) matrices over \(\mathcal{I}\) and the set of all column \(n\)-vectors over \(\mathcal{I}\) by \(\mathcal{I}(m, n)\) and \(\mathcal{I}(n)\), respectively. They consider the ordering \(\leq\) on the sets \(\mathcal{I}(m, n)\) and \(\mathcal{I}(n)\) as follows:
\begin{itemize}
\item[\(\bullet\)] for \(A,C \in \mathcal{I}(m, n): A \leq C\) if \(a_{i j} \leq c_{i j}\) for each \(i \in M\) and for each \(j \in N\)
\item[\(\bullet\)] for \(x, y \in \mathcal{I}(n): x \leq y\) if \(x_{j} \leq y_{j}\) for each \(j \in N\).
\end{itemize}
Hence, they define interval matrices \(\boldsymbol{A}\) as follows:
\[
\boldsymbol{A}=[\underline{A}, \overline{A}]=\{A \in \mathcal{I}(m, n) : \underline{A} \leq A \leq\overline{A}\}
\]
and introduce an interval max-min matrix equation \(\boldsymbol{A} \otimes X \otimes \boldsymbol{C}=\boldsymbol{B}\), where \(\boldsymbol{A}, \boldsymbol{B},\) and \(\boldsymbol{C}\) are given interval matrices. They investigate the solvability of interval matrix equations in max-min algebra and they discuss three types of solvability of interval max-min matrix equations.
Reviewer: Mehdi Mohammadzadeh Karizaki (Torbat Heydarieh)Tracial bounds for multilinear Schur multipliers.https://www.zbmath.org/1459.150232021-05-28T16:06:00+00:00"Skripka, Anna"https://www.zbmath.org/authors/?q=ai:skripka.annaThe author establishes lower and upper bounds for the trace of an \(n\)-linear Schur multiplier \(\mathfrak{M}_{m(n)}\) associated with the symbol \(m(n)\) in the particular case of certain self-adjoint submatrices of its symbol (see Theorem 3.1), and also characterizes the tracial positivity of the \(n\)-linear multiplier on symmetric tuples (see Theorem 3.2 and Theorem 3.3). These results are generalized to the case of a Schur multiplier based on the spectral data of a self-adjoint operator (see Section 4). The paper is well written.
Reviewer: Nicolae Lupa (Timişoara)