Explicit solution of the matrix equation \(AXB-CXD=E\). (English) Zbl 0682.15013

The authors consider the unique solution of the matrix equation (1) \(AXB- CXD=E\) where \(X\in {\mathbb{K}}^{n\times p}\) is an unknown matrix and \(A,C\in {\mathbb{K}}^{m\times n}\), \(B,D\in {\mathbb{K}}^{p\times q}\), \(E\in {\mathbb{K}}^{n\times p}\) \(({\mathbb{K}}={\mathbb{R}}\) or \({\mathbb{C}})\). Note that equation (1) is equivalent to \((2)\quad (\lambda C-A)XB-CX(\lambda B-D)=- E.\)
In Section 2, when \(m=n\), \(p=q\) and B or C is nonsingular, a theoretical approach to the solution of (1) is presented on the basis of the relative Cayley-Hamilton theorem. Under hypothesis (i) the pencils \(\lambda\) C-A and \(\lambda\) B-D are regular, and (ii) the spectra of these pencils have an empty intersection, the unique solution is obtained by means of the inversion of an \(m\times m\) and \(p\times p\) matrix. This paper is strongly influenced by A. Jameson’s work [SIAM J. Appl. Math. 16, 1020-1023 (1968; Zbl 0169.352)].
Section 3 deals with the case of singular pencils. It is shown that if (iii) the pencils \(\lambda\) C-A and \(\lambda\) B-D are right and left invertible, respectively, and (iv) they have disjoint spectra, then the equation (1) has a solution. The proof is based on the reduction to the case of regular pencils by using the same technique applied P. Van Dooren [Lecture Notes Math. 973, 58-73 (1983; Zbl 0517.65022)] to the matrix equation in Y and Z \((\lambda C-A)Y-Z(\lambda B-D)=\lambda F-E.\)
Reviewer: M.Kono


15A24 Matrix equations and identities
15A22 Matrix pencils
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