## 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

### MSC:

 15A24 Matrix equations and identities 15A22 Matrix pencils

### Citations:

Zbl 0169.352; Zbl 0517.65022
Full Text:

### References:

 [1] Chu, K. E., Exclusion theorems for the generalized eigenvalue problem, (Numerical Analysis Rpt. NA/11/85 (1985), Dept of Mathematics, Univ. of Reading) [2] Epton, M. A., Methods for the solution of AXB − $$CXD = E$$ and its application in the numerical solution of implicit ordinary differential equations, BIT, 20, 341-345 (1980) · Zbl 0452.65015 [4] Gantmacher, F. R., Matrix Theory, Vols. 1,2 (1977), Chelsea, New York · Zbl 0085.01001 [5] Rózsa, P., Lineare Matrizengleichungen und Kroneckersche Produkte, Z. Angew. Math. Mech., 58T, 395-397 (1978) · Zbl 0388.15011 [6] Rózsa, P., Linear matrix equations and Kronecker products, Proceedings of the Fourth Symposium on Basic Problems of Numerical Mathematics, 153-162 (1978), Plzen · Zbl 0454.15009 [7] Ma, Er-Chieh, A finite series solution of the matrix equation AX − $$XB = C$$, SIAM J. Appl. Math., 14, 490-495 (1966) · Zbl 0144.27003 [8] Chu, K. E., The Solution of the Matrix Equations AXB − $$CXD = E$$ and (YA − DZ, YC − BZ) = $$(E,F)$$, (Numerical Analysis Rpt. NA/10/85 (1985), Dept. of Mathematics, Univ. of Reading) · Zbl 0631.15006 [9] Lewis, F. L., Adjoint matrix, Bezout theorem, Cayley-Hamilton theorem and Fadeev’s method for the matrix pencil (sE − $$A)$$, Proceedings of the 22nd IEEE Conference on Decision and Control, 1282-1288 (1983) [10] Jameson, A., Solution of the equation AX − $$XB = C$$ by inversion of an $$M$$ X $$M$$ or $$N$$ X $$N$$ matrix, SIAM J. Appl. Math., 16, 1020-1023 (1968) · Zbl 0169.35202 [11] Van Dooren, P., Reducing subspaces: Definitions, properties and algorithms, (Ruche, A.; Kagstrom, B., Matrix Pencils. Matrix Pencils, Lecture Notes in Math. (1983), Springer: Springer New York), 58-73 · Zbl 0517.65022 [12] Mitra, S. K., The matrix equation $$AXB + CXD = E$$, SIAM J. Appl. Math., 32, 823-825 (1977) · Zbl 0392.15005
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