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

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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 $X$.
    Mathematics Subject Classification: Primary: 15A09, 15A24; Secondary: 65K10; 65K15.

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  • [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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