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The solution of the matrix equations \(AXB-CXD=E\) and \((YA-DZ,YC- BZ)=(E,F)\). (English) Zbl 0631.15006
The author proves that a solution of the first equation in the title, as well as the second system, is unique iff (i) pencils A-\(\lambda\) C and D- \(\lambda\) B are regular and (ii) \(\rho (A,C)\cap \rho (B,D)=\phi\) where \(\rho (M,N)=\{(\gamma,\alpha)/\gamma Mx=\alpha Nx\) for some \(x\neq 0\) and \((\gamma,\alpha)\equiv (\delta,\beta)\) iff \(\alpha \delta =\beta \gamma \}\). The author suggests an algoritm to solve the equation that involves transforming (A,C) to low-triangular and (B,D) to upper-triangular Schur form and evaluates the number of operations required to carry out the algorithm. The cases when (i) and/or (ii) above are not satisfied are also studied. It is shown that the system of equations \((YA-DZ,YC- BZ)=(E,F)\) is equivalent to the equation \(AXB-CXD=E\) and an algorithm to solve the system is proposed.
Reviewer: B.Reichstein

MSC:
15A24 Matrix equations and identities
65F30 Other matrix algorithms (MSC2010)
Software:
Algorithm 432
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References:
[1] Bartels, R.H.; Stewart, G.W., Solution of the matrix equation AX+XB=C, Comm. ACM, 15, 820-826, (1972) · Zbl 1372.65121
[2] Chu, K.-W.E., Exclusion theorems for the generalized eigenvalue problem, (), (to appear in SIAM J. Numer. Anal.).
[3] Cline, A.K.; Moler, C.B.; Stewart, G.W.; Wilkinson, J.H., An estimate for the condition number of a matrix, SIAM J. numer. anal., 16, 368-375, (1979) · Zbl 0403.65012
[4] Epton, M.A., Methods for the solution of AXD−BXC=E and its application in the numerical solution of implicit ordinary differential equations, Bit, 20, 341-345, (1980) · Zbl 0452.65015
[5] Gantmacher, F.R., The theory of matrices, II, (1960), Chelsea New York · Zbl 0088.25103
[6] Golub, G.H.; Nash, S.; Van Loan, C., A Hessenberg-Schur method for the matrix problem AX+XB=C, IEEE trans. automat. control, AC-24, 909-913, (1979) · Zbl 0421.65022
[7] Moler, C.B.; Stewart, G.W., An algorithm for the generalized matrix eigenvalue problem ax=λbx, SIAM J. numer. anal., 10, 241-256, (1973) · Zbl 0253.65019
[8] Nashed, M.Z., Generalized inverses and applications, (1976), Academic New York · Zbl 0346.15001
[9] Stewart, G.W., On the sensitivity of the eigenvalue problem ax=λbx, SIAM J. numer. anal., 9, 669-686, (1972) · Zbl 0252.65026
[10] Van Dooren, P., The computation of Kronecker’s canonical form of a singular pencil, Linear algebra appl., 27, 103-140, (1979) · Zbl 0416.65026
[11] Ward, R.C., The combination shift QZ algorithm, SIAM J. numer. anal., 12, 835-852, (1975) · Zbl 0342.65022
[12] Wilkinson, J.H., The algebraic eigenvalue problem, (1965), Clarendon Oxford · Zbl 0258.65037
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