×

zbMATH — the first resource for mathematics

A survey on rank and inertia optimization problems of the matrix-valued function \(A+BXB^\ast\). (English) Zbl 1327.15007
Summary: This paper is concerned with some rank and inertia optimization problems of the Hermitian matrix-valued functions \(A + BXB^{*}\) subject to restrictions. We first establish several groups of explicit formula for calculating the maximum and minimum ranks and inertias of matrix sum \(A + X\) subject to a Hermitian matrix \(X\) that satisfies a fixed-rank and semi-definiteness restrictions by using some discrete and matrix decomposition methods. We then derive formulas for calculating the maximum and minimum ranks and inertias of the matrix-valued function \(A + BXB^*\) subject to a Hermitian matrix \(X\) that satisfies a fixed-rank and semi-definiteness restrictions, and show various properties \(A + BXB^{*}\) from these ranks and inertias formulas. In particular, we give necessary and sufficient conditions for the equality \(A + BXB^* = 0\) and the inequality \(A + BXB^* \succ 0\, (\succeq 0, \prec 0, \, \preceq 0)\) to hold respectively for these specified Hermitian matrices \(X\).

MSC:
15A09 Theory of matrix inversion and generalized inverses
15A24 Matrix equations and identities
65K10 Numerical optimization and variational techniques
65K15 Numerical methods for variational inequalities and related problems
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] W. Ai, On the low rank solutions for linear matrix inequalities,, Math. Oper. Res., 33, 965, (2008) · Zbl 1218.90152
[2] E. M. de Sá, On the inertia of sums of Hermitian matrices,, Linear Algebra Appl., 37, 143, (1981) · Zbl 0457.15012
[3] D. A. Gregory, Inertia and biclique decompositions of joins of graphs,, J. Combin. Theory Ser. B, 88, 135, (2003) · Zbl 1025.05042
[4] M. Journée, Low-rank optimization on the cone of positive semidefinite matrices,, SIAM J. Optim., 20, 2327, (2010) · Zbl 1215.65108
[5] C.-K. Li, Sum of Hermitian matrices with given eigenvalues: inertia, rank, and multiple eigenvalues,, Canad. J. Math., 62, 109, (2010) · Zbl 1188.15039
[6] Y. Liu, More on extremal ranks of the matrix expressions A-BX± X<SUP>*</SUP>B<SUP>*</SUP> with statistical applications,, Numer. Linear Algebra Appl., 15, 307, (2008) · Zbl 1212.15029
[7] Y. Liu, Extremal ranks of submatrices in an Hermitian solution to the matrix equation AXA<SUP>*</SUP>= B with applications,, J. Appl. Math. Comput., 32, 289, (2010) · Zbl 1194.15014
[8] Y. Liu, A simultaneous decomposition of a matrix triplet with applications,, Numer. Linear Algebra Appl., 18, 69, (2011) · Zbl 1249.15020
[9] Y. Liu, Max-min problems on the ranks and inertias of the matrix expressions A-BXC ± (BXC)<sup>*</sup> with applications,, J. Optim. Theory Appl., 148, 593, (2011) · Zbl 1223.90077
[10] Y. Liu, Hermitian-type of singular value decomposition for a pair of matrices and its applications,, Numer. Linear Algebra Appl., 20, 60, (2013) · Zbl 1289.65108
[11] Y. Liu, Ranks of Hermitian and skew-Hermitian solutions to the matrix equation AXA=B<SUP>*</SUP>,, Linear Algebra Appl., 431, 2359, (2009) · Zbl 1180.15018
[12] C. Lu, Revisit to the problem of generalized low rank approximation of matrices,, In: ICIC 2006 (D.-S. Huang, 345, 450, (2006) · Zbl 1202.65057
[13] J. H. Manton, The geometry of weighted low-rank approximations,, IEEE Trans. Sign. Process., 51, 500, (2003) · Zbl 1369.94221
[14] G. Marsaglia, Equalities and inequalities fo ranks of matrices,, Linear Multilinear Algebra, 2, 269, (1974)
[15] D. V. Ouellette, Schur complements and statistics,, Linear Algebra Appl., 36, 187, (1981) · Zbl 0455.15012
[16] R. E. Skelton, <em>A Unified Algebraic Approach to Linear Control Design</em>,, Taylor & Francis, (1997)
[17] Y. Tian, Solvability of two linear matrix equations,, Linear Multilinear Algebra, 48, 123, (2000) · Zbl 0970.15005
[18] Y. Tian, Equalities and inequalities for inertias of Hermitian matrices with applications,, Linear Algebra Appl., 433, 263, (2010) · Zbl 1205.15033
[19] Y. Tian, Rank and inertia of submatrices of the Moore-Penrose inverse of a Hermitian matrix,, Electron. J. Linear Algebra, 20, 226, (2010) · Zbl 1207.15007
[20] Y. Tian, Completing block Hermitian matrices with maximal and minimal ranks and inertias,, Electron. J. Linear Algebra, 21, 124, (2010) · Zbl 1207.15029
[21] Y. Tian, Maximization and minimization of the rank and inertia of the Hermitian matrix expression A - BX - (BX)<sup>*</sup> with applications,, Linear Algebra Appl., 434, 2109, (2011) · Zbl 1211.15022
[22] Y. Tian, Solutions to 18 constrained optimization problems on the rank and inertia of the linear matrix function A + BXB<SUP>*</SUP>,, Math. Comput. Modelling, 55, 955, (2012) · Zbl 1255.15010
[23] Y. Tian, On additive decompositions of the Hermitian solutions of the matrix equation AXA<SUP>*</SUP>= B,, Mediterr. J. Math., 9, 47, (2012) · Zbl 1241.15012
[24] Y. Tian, On an equality and four inequalities for generalized inverses of Hermitian matrices,, Electron. J. Linear Algebra, 23, 11, (2012) · Zbl 1253.15025
[25] Y. Tian, Equalities and inequalities for Hermitian solutions and Hermitian definite solutions of the two matrix equations AX = B and AXA<SUP>*</SUP> = B,, Aequat. Math., 86, 107, (2013) · Zbl 1281.15016
[26] Y. Tian, Some optimization problems on ranks and inertias of matrix-valued functions subject to linear matrix equation restrictions,, Banach J. Math. Anal., 8, 148, (2014) · Zbl 1278.15019
[27] Y. Tian, Extremal ranks of some symmetric matrix expressions with applications,, SIAM J. Matrix Anal. Appl., 28, 890, (2006) · Zbl 1123.15001
[28] J. Ye, Generalized low rank approximations of matrices,, Machine Learning, 61, 167, (2005) · Zbl 1087.65043
[29] H. Zha, A note on the existence of the hyperbolic singular value decomposition,, Linear Algebra Appl., 240, 199, (1996) · Zbl 0923.15007
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.