论文标题
几个矩阵的Schur分解
Schur decomposition of several matrices
论文作者
论文摘要
经常使用并研究了与周期特征值问题相关的单个矩阵，一对矩阵或与矩阵的集合的Schur分解和相应的Schur形式。这些形式是三角形复杂矩阵或准三角形的真实矩阵，它们分别通过单一或正交转换等效于原始矩阵。通常，出于理论和数值目的，我们通常需要通过可接受的转换来减少矩阵的矩阵集合。不幸的是，这种减少并不总是可能的。在本文中，我们描述了可以通过相应的统一（正交）变换将复杂（真实）矩阵的所有集合简化为Schur形式的集合，并解释了如何进行这种减少。我们证明该类包括与伪索图相关的矩阵集合。换句话说，我们描述了何时存在矩阵集合以及如何找到矩阵的何时。
Schur decompositions and the corresponding Schur forms of a single matrix, a pair of matrices, or a collection of matrices associated with the periodic eigenvalue problem are frequently used and studied. These forms are upper-triangular complex matrices or quasi-upper-triangular real matrices that are equivalent to the original matrices via unitary or, respectively, orthogonal transformations. In general, for theoretical and numerical purposes we often need to reduce, by admissible transformations, a collection of matrices to the Schur form. Unfortunately, such a reduction is not always possible. In this paper we describe all collections of complex (real) matrices that can be reduced to the Schur form by the corresponding unitary (orthogonal) transformations and explain how such a reduction can be done. We prove that this class consists of the collections of matrices associated with pseudoforest graphs. In other words, we describe when the Schur form of a collection of matrices exists and how to find it.
