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. Google Scholar

[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. Google Scholar

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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. Google Scholar

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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. Google Scholar

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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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[14]

G. Marsaglia and G. P. H. Styan, Equalities and inequalities fo ranks of matrices,, Linear Multilinear Algebra, 2 (1974), 269. Google Scholar

[15]

D. V. Ouellette, Schur complements and statistics,, Linear Algebra Appl., 36 (1981), 187. doi: 10.1016/0024-3795(81)90232-9. Google Scholar

[16]

R. E. Skelton, T. Iwasaki and K. M. Grigoriadis, A Unified Algebraic Approach to Linear Control Design,, Taylor & Francis, (1997). Google Scholar

[17]

Y. Tian, Solvability of two linear matrix equations,, Linear Multilinear Algebra, 48 (2000), 123. doi: 10.1080/03081080008818664. Google Scholar

[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. Google Scholar

[19]

Y. Tian, Rank and inertia of submatrices of the Moore-Penrose inverse of a Hermitian matrix,, Electron. J. Linear Algebra, 20 (2010), 226. Google Scholar

[20]

Y. Tian, Completing block Hermitian matrices with maximal and minimal ranks and inertias,, Electron. J. Linear Algebra, 21 (2010), 124. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[24]

Y. Tian, On an equality and four inequalities for generalized inverses of Hermitian matrices,, Electron. J. Linear Algebra, 23 (2012), 11. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[28]

J. Ye, Generalized low rank approximations of matrices,, Machine Learning, 61 (2005), 167. Google Scholar

[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. Google Scholar

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. doi: 10.1287/moor.1080.0331. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[14]

G. Marsaglia and G. P. H. Styan, Equalities and inequalities fo ranks of matrices,, Linear Multilinear Algebra, 2 (1974), 269. Google Scholar

[15]

D. V. Ouellette, Schur complements and statistics,, Linear Algebra Appl., 36 (1981), 187. doi: 10.1016/0024-3795(81)90232-9. Google Scholar

[16]

R. E. Skelton, T. Iwasaki and K. M. Grigoriadis, A Unified Algebraic Approach to Linear Control Design,, Taylor & Francis, (1997). Google Scholar

[17]

Y. Tian, Solvability of two linear matrix equations,, Linear Multilinear Algebra, 48 (2000), 123. doi: 10.1080/03081080008818664. Google Scholar

[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. Google Scholar

[19]

Y. Tian, Rank and inertia of submatrices of the Moore-Penrose inverse of a Hermitian matrix,, Electron. J. Linear Algebra, 20 (2010), 226. Google Scholar

[20]

Y. Tian, Completing block Hermitian matrices with maximal and minimal ranks and inertias,, Electron. J. Linear Algebra, 21 (2010), 124. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[24]

Y. Tian, On an equality and four inequalities for generalized inverses of Hermitian matrices,, Electron. J. Linear Algebra, 23 (2012), 11. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[28]

J. Ye, Generalized low rank approximations of matrices,, Machine Learning, 61 (2005), 167. Google Scholar

[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. Google Scholar

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