zbMATH — the first resource for mathematics

Solutions to 18 constrained optimization problems on the rank and inertia of the linear matrix function. (English) Zbl 1255.15010
Summary: The inertia of a Hermitian matrix is defined to be a triplet composed by the numbers of the positive, negative and zero eigenvalues of the matrix counted with multiplicities, respectively. If we take the inertia and rank of a Hermitian matrix as objective functions, then they are neither differentiable nor smooth. In this case, maximizing and minimizing the inertia and rank of a Hermitian matrix function could be regarded as a continuous-integer optimization problem. In this paper, we use some pure algebraic operations of matrices and their generalized inverses to derive explicit expansion formulas for calculating the global maximum and minimum ranks and inertias of the linear Hermitian matrix function \(A+BXB^{\ast }\) subject to some rank and definiteness restrictions on the variable matrix X. Various direct consequences of the formulas in characterizing algebraic properties of \(A+BXB^{\ast }\) are also presented. In particular, solutions to a group of constrained optimization problems on the rank and inertia of a partially specified block Hermitian matrix are given.

15A18 Eigenvalues, singular values, and eigenvectors
Full Text: DOI
[1] Skelton, R.E.; Iwasaki, T.; Grigoriadis, K.M., A unified algebraic approach to linear control design, (1997), Taylor & Francis London
[2] Y. Liu, Y. Tian, Hermitian-type of singular value decomposition with applications, Numer. Linear Algebra Appl. (accepted).
[3] Tian, Y., Equalities and inequalities for inertias of Hermitian matrices with applications, Linear algebra appl., 433, 263-296, (2010) · Zbl 1205.15033
[4] Tian, Y.; Liu, Y., Extremal ranks of some symmetric matrix expressions with applications, SIAM J. matrix anal. appl., 28, 890-905, (2006) · Zbl 1123.15001
[5] Liu, Y.; Tian, Y., Extremal ranks of submatrices in an Hermitian solution to the matrix equation \(A X A^\ast = B\) with applications, J. appl. math. comput., 32, 289-301, (2010) · Zbl 1194.15014
[6] Liu, Y.; Tian, Y.; Takane, Y., Ranks of Hermitian and skew-Hermitian solutions to the matrix equation \(A X A^\ast = B\), Linear algebra appl., 431, 2359-2372, (2009) · Zbl 1180.15018
[7] Tian, Y., Maximization and minimization of the rank and inertia of the Hermitian matrix expression \(A - B X -(B X)^\ast\) with applications, Linear algebra appl., 434, 2109-2139, (2011) · Zbl 1211.15022
[8] Liu, Y.; Tian, Y., MAX-MIN problems on the ranks and inertias of the matrix expressions \(A - B X C \pm(B X C)^\ast\) with applications, J. optim. theory appl., 148, 593-622, (2011) · Zbl 1223.90077
[9] Chu, D.; Hung, Y.S.; Woerdeman, H.J., Inertia and rank characterizations of some matrix expressions, SIAM J. matrix anal. appl., 31, 1187-1226, (2009) · Zbl 1198.15010
[10] Liu, Y.; Tian, Y., More on extremal ranks of the matrix expressions \(A - B X \pm X^\ast B^\ast\) with statistical applications, Numer. linear algebra appl., 15, 307-325, (2008) · Zbl 1212.15029
[11] Liu, Y.; Tian, Y., A simultaneous decomposition of a matrix triplet with applications, Numer. linear algebra appl., 18, 69-85, (2011) · Zbl 1249.15020
[12] Tian, Y., Completing block Hermitian matrices with maximal and minimal ranks and inertias, Electron. linear algebra appl., 21, 124-141, (2010) · Zbl 1207.15029
[13] Y. Tian, On additive decompositions of the Hermitian solutions of the matrix equation \(A X A^\ast = B\), Mediterr. J. Math. doi:10.1007/s00009-010-0110-8.
[14] Ai, W.; Huang, Y.; Zhang, S., On the low rank solutions for linear matrix inequalities, Math. oper. res., 33, 965-975, (2008) · Zbl 1218.90152
[15] Journée, M.; Bach, F.; Absil, P.-A.; Sepulchre, R., Low-rank optimization on the cone of positive semidefinite matrices, SIAM J. optim., 20, 2327-2351, (2010) · Zbl 1215.65108
[16] Lu, C.; Liu, W.; An, S., Revisit to the problem of generalized low rank approximation of matrices, (), 450-460 · Zbl 1202.65057
[17] Manton, J.H.; Mahony, R.; Hua, Y., The geometry of weighted low-rank approximations, IEEE trans. signal process., 51, 500-514, (2003) · Zbl 1369.94221
[18] Brodlie, K.W.; Gourlay, D.A.; Greenstadt, J., Rank-one and rank-two ccorrections to positive definite matrices expressed in product, IMA J. appl. math., 11, 73-82, (1973) · Zbl 0252.15007
[19] Hoang, T.M.; Thierauf, T., The complexity of the inertia, (), 206-217 · Zbl 1027.68063
[20] T.M. Hoang, T. Thierauf, The complexity of the inertia and some closure properties of GapL, in: Proceedings of the Twentieth Annual IEEE Conference on Computational Complexity, pp. 28-37, 2005.
[21] Tian, Y., Rank and inertia of submatrices of the moore – penrose inverse of a Hermitian matrix, Electron. J. linear algebra, 20, 226-240, (2010) · Zbl 1207.15007
[22] Marsaglia, G.; Styan, G.P.H., Equalities and inequalities for ranks of matrices, Linear multilinear algebra, 2, 269-292, (1974)
[23] Ostrowski, A., A quantitative formulation of sylvester’s law of inertia II, Proc. nat. acad. sci. USA, 46, 859-862, (1960) · Zbl 0095.01205
[24] Cohen, N.; Dancis, J., Maximal rank Hermitian completions of partially specified Hermitian matrices, Linear algebra appl., 244, 265-276, (1996) · Zbl 0857.15018
[25] Cohen, N.; Dancis, J., Inertias of block band matrix completions, SIAM J. matrix anal. appl., 19, 583-612, (1998) · Zbl 0974.15009
[26] da Fonseca, C.M., The inertia of certain Hermitian block matrices, Linear algebra appl., 274, 193-210, (1998) · Zbl 0929.15019
[27] da Fonseca, C.M., The inertia of Hermitian block matrices with zero main diagonal, Linear algebra appl., 311, 153-160, (2000) · Zbl 0961.15013
[28] Dancis, J., The possible inertias for a Hermitian matrix and its principal submatrices, Linear algebra appl., 85, 121-151, (1987) · Zbl 0614.15011
[29] Dancis, J., Positive semidefinite completions of partial Hermitian matrices, Linear algebra appl., 175, 97-114, (1992) · Zbl 0760.15019
[30] Gregory, D.A.; Heyink, B.; Vander Meulen, K.N., Inertia and biclique decompositions of joins of graphs, J. combin. theory ser. B, 88, 135-151, (2003) · Zbl 1025.05042
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.