# American Institute of Mathematical Sciences

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 $A + BXB^{*}$

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

Received  April 2014 Revised  July 2015 Published  August 2015

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 $X$.
Citation: Yongge Tian. A survey on rank and inertia optimization problems of the matrix-valued function $A + BXB^{*}$. Numerical Algebra, Control & 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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, 345 (2006), 450. [13] J. H. Manton, R. Mahony and Y. Hua, The geometry of weighted low-rank approximations,, IEEE Trans. Sign. Process., 51 (2003), 500. 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. [15] D. V. Ouellette, Schur complements and statistics,, Linear Algebra Appl., 36 (1981), 187. 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, (1997). [17] Y. Tian, Solvability of two linear matrix equations,, Linear Multilinear Algebra, 48 (2000), 123. doi: 10.1080/03081080008818664. [18] Y. Tian, Equalities and inequalities for inertias of Hermitian matrices with applications,, Linear Algebra Appl., 433 (2010), 263. 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. [20] Y. Tian, Completing block Hermitian matrices with maximal and minimal ranks and inertias,, Electron. J. Linear Algebra, 21 (2010), 124. [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. 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. 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. 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. [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. 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. [27] Y. Tian and Y. Liu, Extremal ranks of some symmetric matrix expressions with applications,, SIAM J. Matrix Anal. Appl., 28 (2006), 890. doi: 10.1137/S0895479802415545. [28] J. Ye, Generalized low rank approximations of matrices,, Machine Learning, 61 (2005), 167. [29] H. Zha, A note on the existence of the hyperbolic singular value decomposition,, Linear Algebra Appl., 240 (1996), 199. doi: 10.1016/0024-3795(94)00197-9.

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##### References:
 [1] W. Ai, Y. Huang and S. Zhang, On the low rank solutions for linear matrix inequalities,, Math. Oper. Res., 33 (2008), 965. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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, 345 (2006), 450. [13] J. H. Manton, R. Mahony and Y. Hua, The geometry of weighted low-rank approximations,, IEEE Trans. Sign. Process., 51 (2003), 500. 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. [15] D. V. Ouellette, Schur complements and statistics,, Linear Algebra Appl., 36 (1981), 187. 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, (1997). [17] Y. Tian, Solvability of two linear matrix equations,, Linear Multilinear Algebra, 48 (2000), 123. doi: 10.1080/03081080008818664. [18] Y. Tian, Equalities and inequalities for inertias of Hermitian matrices with applications,, Linear Algebra Appl., 433 (2010), 263. 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. [20] Y. Tian, Completing block Hermitian matrices with maximal and minimal ranks and inertias,, Electron. J. Linear Algebra, 21 (2010), 124. [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. 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. 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. 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. [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. 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. [27] Y. Tian and Y. Liu, Extremal ranks of some symmetric matrix expressions with applications,, SIAM J. Matrix Anal. Appl., 28 (2006), 890. doi: 10.1137/S0895479802415545. [28] J. Ye, Generalized low rank approximations of matrices,, Machine Learning, 61 (2005), 167. [29] H. Zha, A note on the existence of the hyperbolic singular value decomposition,, Linear Algebra Appl., 240 (1996), 199. doi: 10.1016/0024-3795(94)00197-9.
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