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A simultaneous decomposition of a matrix triplet with applications. (English) Zbl 1249.15020
The paper concentrates on a matrix expression $$A-BXB^{*}-CYC^{*}$$, where $$A$$ is a given square Hermitian or skew-Hermitian matrix, $$B, C$$ are given (generally rectangular) matrices and $$X, Y$$ are variable matrices. First, motivated by the generalized singular value decomposition of two matrices a simultaneous decomposition of the matrix triplet $$(A,B,C)$$ is derived. Using this result, closed formulas for the maximal and minimal possible ranks of the matrix $$A-BXB^{*}-CYC^{*}$$ are given under the assumption $$X = \pm X^{*}, Y = \pm Y^{*}$$. Moreover, necessary and sufficient conditions for existence of two Hermitian or skew-Hermitian solutions of the equation $$A=BXB^{*}+CYC^{*}$$ are derived. These conditions can be simply verified and involve ranks of the individual matrices $$A,B,C$$. Formulas for the solutions are also presented. Finally, some further open problems related to the considered expression $$A-BXB^{*}-CYC^{*}$$ are summarized.

MSC:
 15A24 Matrix equations and identities 15A23 Factorization of matrices 15A03 Vector spaces, linear dependence, rank, lineability 15B57 Hermitian, skew-Hermitian, and related matrices
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References:
 [1] Liu, More on extremal ranks of the matrix expressions A-BX{$$\pm$$}X*B* with statistical applications, Numerical Linear Algebra with Applications 15 pp 307– (2008) · Zbl 1212.15029 [2] Tian, Extremal ranks of some symmetric matrix expressions with applications, SIAM Journal on Matrix Analysis and Applications 28 pp 890– (2006) · Zbl 1123.15001 [3] Liu, Extremal ranks of submatrices in an Hermitian solutions to the matrix equation AXA*=B with applications, Journal of Applied Mathematics and Computing · Zbl 1194.15014 [4] Liu, Ranks of Hermitian and skew-Hermitian solutions to the matrix equation AXA*=B, Linear Algebra and its Applications 431 pp 2359– (2009) · Zbl 1180.15018 [5] Tian, The inverse of any two-by-two nonsingular partitioned matrix and three matrix inverse completion problems, Computers and Mathematics with Applications 57 pp 1294– (2009) · Zbl 1186.15005 [6] Marsaglia, Equalities and inequalities for ranks of matrices, Linear and Multilinear Algebra 2 pp 269– (1974) · Zbl 0297.15003 [7] Tian, Sum decompositions of the OLSE and BLUE under a partitioned linear model, International Statistical Review 75 pp 224– (2007) [8] Paige, Towards a generalized singular value decomposition, SIAM Journal on Numerical Analysis 18 pp 398– (1981) · Zbl 0471.65018 [9] Zha, The restricted singular value decomposition of matrix triplets, SIAM Journal on Matrix Analysis and Applications 12 pp 172– (1991) · Zbl 0722.15011 [10] Zha, The product-product singular value decomposition of matrix triplets, BIT 31 pp 711– (1991) · Zbl 0743.65040 [11] De Moor, A tree of generalizations of the ordinary singular value decomposition, Linear Algebra and its Applications 147 pp 469– (1991) · Zbl 0715.15006 [12] Liu, A mixed-type reverse order law for generalized inverse of a triple matrix product (in Chinese), Acta Mathematica Sinica 52 pp 197– (2009) · Zbl 1199.15021 [13] Liu, On the block independence in g-inverse and reflexive inner inverse of a partitioned matrix, Acta Mathematica Sinica, English Series 23 pp 723– (2007) · Zbl 1123.15003 [14] Hong, On simultaneous reduction of families of matrices to triangular or diagonal form by unitary congruence, Linear and Multilinear Algebra 17 pp 271– (1985) · Zbl 0568.15007 [15] Berman, Mass matrix correction using an incomplete set of measure model, AIAA Journal 17 pp 1147– (1979) [16] Berman, Improvement of a large analytical model using test data, AIAA Journal 21 pp 1168– (1983) [17] Baksalary, Nonnegative definite and positive definite solutions to the matrix equation AXA*=B, Linear and Multilinear Algebra 16 pp 133– (1984) · Zbl 0552.15009 [18] Chang, The symmetric solutions of the matrix equations AX + YB=C, AXAT + BYBT=C and (ATXA, BTXB)=(C, D), Linear Algebra and its Applications 179 pp 171– (1993) · Zbl 0765.15002 [19] Deng, On solutions of matrix equation AXAT + BYBT=C, Journal of Computational Mathematics 23 pp 17– (2005) [20] Groß, Nonnegtive-definite and positive-definite solution to the matrix equation AXA*=B -revisited, Linear Algebra and its Applications 321 pp 123– (2000) · Zbl 0984.15011 [21] Khatri, Hermitian and nonnegative definite solutions of linear matrix equations, SIAM Journal on Applied Mathematics 31 pp 579– (1976) · Zbl 0359.65033 [22] Liao, The constrained solutions of two matrix equations, Acta Mathematica Sinica, English Series 18 pp 671– (2002) · Zbl 1028.15011 [23] Wei, On rank-constrained Hermitian nonnegtive-definite least squares solutions to the matrix equation AXA*=B, International Journal of Computer Mathematics 84 pp 945– (2007) · Zbl 1129.15012 [24] Xu, On solutions of matrix equation AXB + CYD=F, Linear Algebra and its Applications 279 pp 93– (1998) · Zbl 0933.15024 [25] Zhang, The general Hermitian nonnegative-definite and positive-definite solutions to the matrix equation GXG* + HYH *=C, Journal of Applied Mathematics and Computing 14 pp 51– (2004) · Zbl 1042.15010 [26] Zhang, The rank-constrained hermitian nonnegtive-definite and positive-definite solutions to the matrix equation AXA*=B, Linear Algebra and its Applications 370 pp 163– (2003) · Zbl 1026.15011 [27] Johnson, Proceedings of Symposia of Applied Mathematics 40 pp 171– (1990) [28] Barrett, Completing a block diagonal matrix with a partially prescribed inverse, Linear Algebra and its Applications 223/4224 pp 73– (1995) · Zbl 0831.15010 [29] Dai, Completing a symmetric 2 {$$\times$$} 2 block matrix and its inverse, Linear Algebra and its Applications 235 pp 235– (1996) · Zbl 0848.15008 [30] Geelen, Maximum rank matrix completion, Linear Algebra and its Applications 288 pp 211– (1999) · Zbl 0933.15026 [31] Harvey, Proceedings of the Seventeenth Annual ACM-SIAM Symposium on Discrete Algorithm pp 1103– (2006) · Zbl 1192.68322 [32] Laurent, The Encyclopedia of Optimization 3 pp 221– (2001) [33] Mahajan, Lecture Notes in Computer Science 4649 pp 269– (2007)
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