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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)
##### Keywords:
matrix equation; pencils; least-squares-type solution
Algorithm 432
Full Text:
##### References:
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