×

Rank equalities related to the generalized inverse \(A^{(2)}_{T,S}\) with applications. (English) Zbl 1159.15006

The authors establish the rank equalities of some matrix expressions and certain block matrices related to the generalized inverse \(A_{T,S}^{(2)}\). They define the block independence in the generalized inverse \(A_{T,S}^{(2)}\) and derive necessary and sufficient conditions for two, three and four ordered matrices to be independent in \(A_{T,S}^{(2)}\), respectively. As special cases, they present the corresponding results on the weighted Moore-Penrose inverse and the Drazin inverse. See Y.-H. Liu and M.-S. Wei [Acta Math. Sin., Engl. Ser. 23, No. 4, 723–730 (2007; Zbl 1123.15003)] and Y. Wang [SIAM J. Matrix Anal. Appl. 19, No. 2, 407–415 (1998; Zbl 0926.15003)] as some related materials.

MSC:

15A09 Theory of matrix inversion and generalized inverses
15A18 Eigenvalues, singular values, and eigenvectors
15A03 Vector spaces, linear dependence, rank, lineability
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Ben-Israel, A.; Greville, T. N.E., Generalized Inverses: Theory and Applications (1974), Wiley: Wiley New York · Zbl 0305.15001
[2] Wei, Y., A characterization and representation of the generalized inverse and its applications, Linear Algebra Appl., 280, 87-96 (1998) · Zbl 0934.15003
[3] Wang, G. W.; Zheng, B., The reverse order law for the generalized inverse \(A_{T, S}^{(2)}\), Appl. Math. Comput., 157, 295-305 (2004) · Zbl 1089.15007
[4] Liu, Y.; Wei, M., Rank equalities related to the generalized inverses \(A_{R, T}^{(2)}, B_{R_1, T_1}^{(2)}\) of two matrices \(A\) and \(B\), Appl. Math. Comput., 159, 19-28 (2004)
[5] Liu, Y.; Wei, M., Rank equalities for submatrices in generalized inverse \(M_{R, T}^{(2)}\) of \(M\), Appl. Math. Comput., 152, 499-504 (2004) · Zbl 1054.15002
[6] Zhou, J. H.; Wang, G., Block idempotent matrices and generalized Schur complement, Appl. Math. Comput., 188, 246-256 (2007) · Zbl 1119.15007
[7] Hall, F. J., Generalized inverse of a bordered matrix of operators, SIAM J. Appl. Math., 29, 152-163 (1975) · Zbl 0324.47007
[8] Hall, F. J., On the independence of blocks of generalized inverses of bordered matrices, Linear Algebra Appl., 14, 53-61 (1976) · Zbl 0345.15004
[9] Hall, F. J.; Hartwig, R. E., Further result on generalized inverses of partitioned matrices, SIAM J. Appl. Math., 30, 617-624 (1976) · Zbl 0345.15005
[10] Wang, Y., On the block independence in the reflexive inner inverse and M-P inverse of block matrix, SIAM J. Matrix Anal. Appl., 19, 2, 407-415 (1998) · Zbl 0926.15003
[11] Liu, Y.; Wei, M., On the block independence in \(G\)-inverse and reflexive inner inverse of a block matrix, Acta Math. Sin., Eng. Ser., 23, 4, 723-730 (2007) · Zbl 1123.15003
[12] Marsaglia, G.; Styan, G. P.H., Equalities and inequalities for ranks of matrices, Linear Multilinear Algebra, 2, 269-292 (1974) · Zbl 0297.15003
[13] Tian, Y., Rank equalities for block matrices and their Moore-Penrose inverses, Houston J. Math., 30, 2, 483-510 (2004) · Zbl 1057.15006
[14] Tian, Y., How to characterize equalities for the Moore-Penrose inverse of a matrix?, Kyungpook Math. J., 41, 1-15 (2001) · Zbl 0987.15001
[15] Zhang, F., Quaternions and matrices of quaternions, Linear Algebra Appl., 251, 21-57 (1997) · Zbl 0873.15008
[16] Farenick, D. R.; Pidkowich, B. A.F., The spectral theorem in quaternions, Linear Algebra Appl., 371, 75-102 (2003) · Zbl 1030.15015
[17] Zhang, F., Geršgorin type theorems for quaternionic matrices, Linear Algebra Appl., 424, 139-153 (2007) · Zbl 1117.15017
[18] Wang, Q. W.; Wu, Z. C.; Lin, C. Y., Extremal ranks of a quaternion matrix expression subject to consistent systems of quaternion matrix equations with applications, Appl. Math. Comput., 182, 1755-1764 (2006) · Zbl 1108.15014
[19] Wang, Q. W.; Song, G. J.; Lin, C. Y., Extreme ranks of the solution to a consistent system of linear quaternion matrix equations with an application, Appl. Math. Comput., 189, 1517-1532 (2007) · Zbl 1124.15010
[20] Wang, Q. W.; Yu, S. W.; Lin, C. Y., Extreme ranks of a linear quaternion matrix expression subject to triple quaternion matrix equations with applications, Appl. Math. Comput., 195, 733-744 (2008) · Zbl 1149.15012
[21] Wang, Q. W., The general solution to a system of real quaternion matrix equations, Comput. Math. Appl., 49, 665-675 (2005) · Zbl 1138.15004
[22] Wang, Q. W., A system of matrix equations and a linear matrix equation over arbitrary regular rings with identity, Linear Algebra Appl., 384, 43-54 (2004) · Zbl 1058.15015
[23] Wang, Q. W., A system of four matrix equations over von Neumann regular rings and its applications, Acta Math. Sin., Eng. Ser., 21, 2, 323-334 (2005) · Zbl 1083.15021
[24] Wang, Q. W.; Qin, F.; Lin, C. Y., The common solution to matrix equations over a regular ring with applications, India J. Pure Appl. Math., 36, 655-672 (2005) · Zbl 1104.15014
[25] Wang, Q. W.; Chang, H. X.; Lin, C. Y., P-(skew)symmetric common solutions to a pair of quaternion matrix equations, Appl. Math. Comput., 195, 721-732 (2008) · Zbl 1149.15011
[26] Sangwine, S. J.; LE Bihan, N., Quaternion singular value decomposition based on bidiagonalization to a real or complex matrix using quaternion Householder transformations, Appl. Math. Comput., 182, 1, 727-738 (2006) · Zbl 1109.65037
[27] Wang, Q. W.; Zhang, F., The reflexive re-nonnegative definite solution to a quaternion matrix equation, Electron. J. Linear Algebra, 17, 88-101 (2008) · Zbl 1147.15012
[28] Wang, Q. W.; Zhang, H. S., On solutions to the quaternion matrix equation \(AXB + CYD = E\), Electron. J. Linear Algebra, 17, 343-358 (2008) · Zbl 1154.15019
[29] Wang, Q. W.; Li, C. K., Ranks and the least-norm of the general solution to a system of quaternion matrix equations, Linear Algebra Appl. (2008)
[30] Yuan, S.; Lao, A. P.; Lei, Y., Least squares Hermitian solution of the matrix equation \((AXB, CXD) = (E, F)\) with the least norm over the skew field of quaternions, Math. Comput. Model., 48, 91-100 (2008) · Zbl 1145.15303
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.