Coupled Sylvester-type Matrix Equations and Block Diagonalization
Coupled Sylvester-type Matrix Equations and Block Diagonalization
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DOI：
10.1137/151005907
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发表时间：
2015-05
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影响因子：
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通讯作者：
Andrii Dmytryshyn;B. Kågström
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文献类型：
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作者：
Andrii Dmytryshyn;B. Kågström
我们证明了Roth型定理的矩阵方程组，包括任意混合的西尔维斯特和$\星星$-西尔维斯特方程，其中的转置或共轭转置的未知矩阵也出现。在充分的一般性，我们推导出一致性条件，证明这样的系统有一个解决方案，当且仅当相关的一组2\倍2$块矩阵表示的方程是块对角化（链接）等价变换。各种应用程序导致几个特定的情况下已经在文献中进行了调查，一些最近和一些很久以前。这些情况的可解性直接来自我们的一般一致性理论。我们还展示了如何将我们的主要结果应用到系统的斯坦型矩阵方程。
We prove Roth-type theorems for systems of matrix equations including an arbitrary mix of Sylvester and $\star$-Sylvester equations, in which the transpose or conjugate transpose of the unknown matrices also 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 $2 \times 2$ 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.