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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
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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 (YADZ, YCBZ)=\((E, F)\), (Numer. Anal. Rpt. NA/10/85 (1985), Dept. of Mathematics, Univ. of Reading: Dept. of Mathematics, Univ. of Reading U.K) · Zbl 0631.15006
[4] Golub, G. H.; Van Loan, C. F., Matrix Computations (1983), Johns Hopkins U.P: Johns Hopkins U.P Baltimore · Zbl 0559.65011
[5] Kolka, G. K.G., Linear matrix equations and pole assignment, (Ph. D. Thesis (1984), Dept. of Mathematics, and Computer Science, Univ. of Salford: Dept. of Mathematics, and Computer Science, Univ. of Salford U.K) · Zbl 0592.93023
[6] Mitra, S. K., Common solutions to a pair of linear matrix equations \(A_1 XB_1=C_1\) and \(A_2 XB_2=C_2\), (Math. Proc. Cambridge, 74 (1973), Philos. Soc), 213-216
[7] Nashed, M. Z., Generalized Inverses and Applications (1976), Academic: 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
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