A survey on rank and inertia optimization problems of the matrix-valued function A + BXB^{*}

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2015, 5(3): 289-326. doi: 10.3934/naco.2015.5.289

A survey on rank and inertia optimization problems of the matrix-valued function $A + BXB^{*}$

Yongge Tian ^1,

1.	CEMA, Central University of Finance and Economics, Beijing 100081, China

Received April 2014 Revised July 2015 Published August 2015

Abstract
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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 $X$.

Keywords: rank, Matrix-valued function, inertia, Löwner partial ordering., optimization, equality, Moore--Penrose inverse, inequality.

Mathematics Subject Classification: Primary: 15A09, 15A24; Secondary: 65K10; 65K1.

Citation: Yongge Tian. A survey on rank and inertia optimization problems of the matrix-valued function $A + BXB^{*}$. Numerical Algebra, Control and Optimization, 2015, 5 (3) : 289-326. doi: 10.3934/naco.2015.5.289

References:

[1]	W. Ai, Y. Huang and S. Zhang, On the low rank solutions for linear matrix inequalities, Math. Oper. Res., 33 (2008), 965-975. doi: 10.1287/moor.1080.0331.
[2]	E. M. de Sá, On the inertia of sums of Hermitian matrices, Linear Algebra Appl., 37 (1981), 143-159. doi: 10.1016/0024-3795(81)90174-9.
[3]	D. A. Gregory, B. Heyink and K. N. Vander Meulen, Inertia and biclique decompositions of joins of graphs, J. Combin. Theory Ser. B, 88 (2003), 135-151. doi: 10.1016/S0095-8956(02)00041-2.
[4]	M. Journée, F. Bach, P.-A. Absil and R. Sepulchre, Low-rank optimization on the cone of positive semidefinite matrices, SIAM J. Optim., 20 (2010), 2327-2351. doi: 10.1137/080731359.
[5]	C.-K. Li and Y.-T. Poon, Sum of Hermitian matrices with given eigenvalues: inertia, rank, and multiple eigenvalues, Canad. J. Math., 62 (2010), 109-132. doi: 10.4153/CJM-2010-007-2.
[6]	Y. Liu and Y. Tian, More on extremal ranks of the matrix expressions A-BX± X^B^ with statistical applications, Numer. Linear Algebra Appl., 15 (2008), 307-325. doi: 10.1002/nla.553.
[7]	Y. Liu and Y. Tian, Extremal ranks of submatrices in an Hermitian solution to the matrix equation AXA^= B with applications, J. Appl. Math. Comput.*, 32 (2010), 289-301. doi: 10.1007/s12190-009-0251-8.
[8]	Y. Liu and Y. Tian, A simultaneous decomposition of a matrix triplet with applications, Numer. Linear Algebra Appl., 18 (2011), 69-85. doi: 10.1002/nla.701.
[9]	Y. Liu and Y. Tian, Max-min problems on the ranks and inertias of the matrix expressions A-BXC ± (BXC)^* with applications, J. Optim. Theory Appl., 148 (2011), 593-622. doi: 10.1007/s10957-010-9760-8.
[10]	Y. Liu and Y. Tian, Hermitian-type of singular value decomposition for a pair of matrices and its applications, Numer. Linear Algebra Appl., 20 (2013), 60-73. doi: 10.1002/nla.1825.
[11]	Y. Liu, Y. Tian and Y. Takane, Ranks of Hermitian and skew-Hermitian solutions to the matrix equation AXA=B^, Linear Algebra Appl., 431* (2009), 2359-2372. doi: 10.1016/j.laa.2009.03.011.
[12]	C. Lu, W. Liu and S. An, Revisit to the problem of generalized low rank approximation of matrices, In: ICIC 2006 (D.-S. Huang, K. Li, and G.W. Irwin, Eds.), LNCIS, 345 (2006), 450-460.
[13]	J. H. Manton, R. Mahony and Y. Hua, The geometry of weighted low-rank approximations, IEEE Trans. Sign. Process., 51 (2003), 500-514. doi: 10.1109/TSP.2002.807002.
[14]	G. Marsaglia and G. P. H. Styan, Equalities and inequalities fo ranks of matrices, Linear Multilinear Algebra, 2 (1974), 269-292.
[15]	D. V. Ouellette, Schur complements and statistics, Linear Algebra Appl., 36 (1981), 187-295. doi: 10.1016/0024-3795(81)90232-9.
[16]	R. E. Skelton, T. Iwasaki and K. M. Grigoriadis, A Unified Algebraic Approach to Linear Control Design, Taylor & Francis, London, 1997.
[17]	Y. Tian, Solvability of two linear matrix equations, Linear Multilinear Algebra, 48 (2000), 123-147. doi: 10.1080/03081080008818664.
[18]	Y. Tian, Equalities and inequalities for inertias of Hermitian matrices with applications, Linear Algebra Appl., 433 (2010), 263-296. doi: 10.1016/j.laa.2010.02.018.
[19]	Y. Tian, Rank and inertia of submatrices of the Moore-Penrose inverse of a Hermitian matrix, Electron. J. Linear Algebra, 20 (2010), 226-240.
[20]	Y. Tian, Completing block Hermitian matrices with maximal and minimal ranks and inertias, Electron. J. Linear Algebra, 21 (2010), 124-141.
[21]	Y. Tian, Maximization and minimization of the rank and inertia of the Hermitian matrix expression A - BX - (BX)^* with applications, Linear Algebra Appl., 434 (2011), 2109-2139. doi: 10.1016/j.laa.2010.12.010.
[22]	Y. Tian, Solutions to 18 constrained optimization problems on the rank and inertia of the linear matrix function A + BXB^, Math. Comput. Modelling*, 55 (2012), 955-968. doi: 10.1016/j.mcm.2011.09.022.
[23]	Y. Tian, On additive decompositions of the Hermitian solutions of the matrix equation AXA^= B, Mediterr. J. Math.*, 9 (2012), 47-60. doi: 10.1007/s00009-010-0110-8.
[24]	Y. Tian, On an equality and four inequalities for generalized inverses of Hermitian matrices, Electron. J. Linear Algebra, 23 (2012), 11-42.
[25]	Y. Tian, Equalities and inequalities for Hermitian solutions and Hermitian definite solutions of the two matrix equations AX = B and AXA^* = B, Aequat. Math., 86 (2013), 107-135. doi: 10.1007/s00010-012-0179-1.
[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 (2014), 148-178.
[27]	Y. Tian and Y. Liu, Extremal ranks of some symmetric matrix expressions with applications, SIAM J. Matrix Anal. Appl., 28 (2006), 890-905. doi: 10.1137/S0895479802415545.
[28]	J. Ye, Generalized low rank approximations of matrices, Machine Learning, 61 (2005), 167-191.
[29]	H. Zha, A note on the existence of the hyperbolic singular value decomposition, Linear Algebra Appl., 240 (1996), 199-205. doi: 10.1016/0024-3795(94)00197-9.

show all references

References:

[1]	W. Ai, Y. Huang and S. Zhang, On the low rank solutions for linear matrix inequalities, Math. Oper. Res., 33 (2008), 965-975. doi: 10.1287/moor.1080.0331.
[2]	E. M. de Sá, On the inertia of sums of Hermitian matrices, Linear Algebra Appl., 37 (1981), 143-159. doi: 10.1016/0024-3795(81)90174-9.
[3]	D. A. Gregory, B. Heyink and K. N. Vander Meulen, Inertia and biclique decompositions of joins of graphs, J. Combin. Theory Ser. B, 88 (2003), 135-151. doi: 10.1016/S0095-8956(02)00041-2.
[4]	M. Journée, F. Bach, P.-A. Absil and R. Sepulchre, Low-rank optimization on the cone of positive semidefinite matrices, SIAM J. Optim., 20 (2010), 2327-2351. doi: 10.1137/080731359.
[5]	C.-K. Li and Y.-T. Poon, Sum of Hermitian matrices with given eigenvalues: inertia, rank, and multiple eigenvalues, Canad. J. Math., 62 (2010), 109-132. doi: 10.4153/CJM-2010-007-2.
[6]	Y. Liu and Y. Tian, More on extremal ranks of the matrix expressions A-BX± X^B^ with statistical applications, Numer. Linear Algebra Appl., 15 (2008), 307-325. doi: 10.1002/nla.553.
[7]	Y. Liu and Y. Tian, Extremal ranks of submatrices in an Hermitian solution to the matrix equation AXA^= B with applications, J. Appl. Math. Comput.*, 32 (2010), 289-301. doi: 10.1007/s12190-009-0251-8.
[8]	Y. Liu and Y. Tian, A simultaneous decomposition of a matrix triplet with applications, Numer. Linear Algebra Appl., 18 (2011), 69-85. doi: 10.1002/nla.701.
[9]	Y. Liu and Y. Tian, Max-min problems on the ranks and inertias of the matrix expressions A-BXC ± (BXC)^* with applications, J. Optim. Theory Appl., 148 (2011), 593-622. doi: 10.1007/s10957-010-9760-8.
[10]	Y. Liu and Y. Tian, Hermitian-type of singular value decomposition for a pair of matrices and its applications, Numer. Linear Algebra Appl., 20 (2013), 60-73. doi: 10.1002/nla.1825.
[11]	Y. Liu, Y. Tian and Y. Takane, Ranks of Hermitian and skew-Hermitian solutions to the matrix equation AXA=B^, Linear Algebra Appl., 431* (2009), 2359-2372. doi: 10.1016/j.laa.2009.03.011.
[12]	C. Lu, W. Liu and S. An, Revisit to the problem of generalized low rank approximation of matrices, In: ICIC 2006 (D.-S. Huang, K. Li, and G.W. Irwin, Eds.), LNCIS, 345 (2006), 450-460.
[13]	J. H. Manton, R. Mahony and Y. Hua, The geometry of weighted low-rank approximations, IEEE Trans. Sign. Process., 51 (2003), 500-514. doi: 10.1109/TSP.2002.807002.
[14]	G. Marsaglia and G. P. H. Styan, Equalities and inequalities fo ranks of matrices, Linear Multilinear Algebra, 2 (1974), 269-292.
[15]	D. V. Ouellette, Schur complements and statistics, Linear Algebra Appl., 36 (1981), 187-295. doi: 10.1016/0024-3795(81)90232-9.
[16]	R. E. Skelton, T. Iwasaki and K. M. Grigoriadis, A Unified Algebraic Approach to Linear Control Design, Taylor & Francis, London, 1997.
[17]	Y. Tian, Solvability of two linear matrix equations, Linear Multilinear Algebra, 48 (2000), 123-147. doi: 10.1080/03081080008818664.
[18]	Y. Tian, Equalities and inequalities for inertias of Hermitian matrices with applications, Linear Algebra Appl., 433 (2010), 263-296. doi: 10.1016/j.laa.2010.02.018.
[19]	Y. Tian, Rank and inertia of submatrices of the Moore-Penrose inverse of a Hermitian matrix, Electron. J. Linear Algebra, 20 (2010), 226-240.
[20]	Y. Tian, Completing block Hermitian matrices with maximal and minimal ranks and inertias, Electron. J. Linear Algebra, 21 (2010), 124-141.
[21]	Y. Tian, Maximization and minimization of the rank and inertia of the Hermitian matrix expression A - BX - (BX)^* with applications, Linear Algebra Appl., 434 (2011), 2109-2139. doi: 10.1016/j.laa.2010.12.010.
[22]	Y. Tian, Solutions to 18 constrained optimization problems on the rank and inertia of the linear matrix function A + BXB^, Math. Comput. Modelling*, 55 (2012), 955-968. doi: 10.1016/j.mcm.2011.09.022.
[23]	Y. Tian, On additive decompositions of the Hermitian solutions of the matrix equation AXA^= B, Mediterr. J. Math.*, 9 (2012), 47-60. doi: 10.1007/s00009-010-0110-8.
[24]	Y. Tian, On an equality and four inequalities for generalized inverses of Hermitian matrices, Electron. J. Linear Algebra, 23 (2012), 11-42.
[25]	Y. Tian, Equalities and inequalities for Hermitian solutions and Hermitian definite solutions of the two matrix equations AX = B and AXA^* = B, Aequat. Math., 86 (2013), 107-135. doi: 10.1007/s00010-012-0179-1.
[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 (2014), 148-178.
[27]	Y. Tian and Y. Liu, Extremal ranks of some symmetric matrix expressions with applications, SIAM J. Matrix Anal. Appl., 28 (2006), 890-905. doi: 10.1137/S0895479802415545.
[28]	J. Ye, Generalized low rank approximations of matrices, Machine Learning, 61 (2005), 167-191.
[29]	H. Zha, A note on the existence of the hyperbolic singular value decomposition, Linear Algebra Appl., 240 (1996), 199-205. doi: 10.1016/0024-3795(94)00197-9.

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