×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite
Full Text: DOI
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)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.