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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

15A24 Matrix equations and identities
65F30 Other matrix algorithms (MSC2010)
Algorithm 432
Full Text: DOI
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