×

zbMATH — the first resource for mathematics

Singular value and generalized singular value decompositions and the solution of linear matrix equations. (English) Zbl 0612.15003
All matrices are taken to be real, but not necessarily square. The author employs the singular value decomposition and a generalization of it to study the solvability of the equation \(AXB+CYD=E\), and of the pair of equations \(AXB=E\), \(FXG=H\). A couple of special cases are considered, and numerical algorithms for the solutions are suggested.
Reviewer: G.P.Barker

MSC:
15A18 Eigenvalues, singular values, and eigenvectors
15A24 Matrix equations and identities
15A09 Theory of matrix inversion and generalized inverses
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Baksalary, J.K.; Kala, R., The matrix equation AX−YB=C, Linear algebra appl., 25, 41-43, (1979) · Zbl 0403.15010
[2] Baksalary, J.K.; Kala, R., The matrix equation AXB−CYD=E, Linear algebra appl., 30, 141-147, (1980) · Zbl 0437.15005
[3] Chu, K.-W.E., The solution of the matrix equations AXB+CXD=E and (YA−DZ, YC−BZ)=(E, F), () · Zbl 0631.15006
[4] Golub, G.H.; Van Loan, C.F., Matrix computations, (1983), Johns Hopkins U.P Baltimore · Zbl 0559.65011
[5] Kolka, G.K.G., Linear matrix equations and pole assignment, () · Zbl 0592.93023
[6] Mitra, S.K., Common solutions to a pair of linear matrix equations A1XB1=C1 and A2XB2=C2, (), 213-216
[7] Nashed, M.Z., Generalized inverses and applications, (1976), Academic New York · Zbl 0346.15001
[8] Paige, C.C.; Saunders, M.A., Towards a generalized singular value decomposition, SIAM J. numer. anal., 18, 398-405, (1981) · Zbl 0471.65018
[9] Roth, W.E., The equations AX−YB=C and AX−XB=C in matrices, Proc. amer. math. soc., 3, 392-396, (1952) · Zbl 0047.01901
[10] Stewart, G.W., Computing the CS-decomposition of a partitioned orthogonal matrix, Numer. math., 40, 297-306, (1982) · Zbl 0516.65016
[11] Zietak, K., The l_p-solution of the linear matrix equation AX+YB=C, Computing, 32, 153-162, (1984) · Zbl 0518.41022
[12] Zietal, K., The Chebyshev solution of the linear matrix equation AX+YB=C, Numer. math., 46, 455-478, (1985) · Zbl 0557.65024
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.