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UMINF 15.04

Coupled Sylvester-type Matrix Equations and Block Diagonalization

We prove Roth’s type theorems for systems of matrix equations including an arbitrary mix of Sylvester and *-Sylvester equations, in which also the transpose or conjugate transpose of the unknown matrices appear. In full generality, we derive consistency conditions by proving that such a system has a solution if and only if the associated set of 2 ×2 block matrix representations of the equations are block diagonalizable by (linked) equivalence transformations. Various applications leading to several particular cases have already been investigated in the literature, some recently and some long ago. Solvability of these cases follow immediately from our general consistency theory. We also show how to apply our main result to systems of Stein-type matrix equations.


matrix equation, Sylvester equation, Stein equation, Roth’s theorem, consistency, block diagonalization


Andrii Dmytryshyn and Bo Kågström

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Entry responsible: Andrii Dmytryshyn

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