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.

翻译：Schur分解及其对应的Schur形式——无论是针对单一矩阵、矩阵对，还是与周期特征值问题相关的矩阵集合——均得到了广泛研究与应用。这些形式或为上三角复矩阵，或为准上三角实矩阵，并通过酉变换（或正交变换）与原矩阵保持等价。一般而言，出于理论分析或数值计算的考虑，我们常需通过允许的变换将矩阵集合约化为Schur形式。然而，这种约化并非总是可行。本文描述了所有可通过相应酉（或正交）变换约化为Schur形式的复（或实）矩阵集合，并阐释了如何实现这种约化。我们证明此类集合恰为与伪森林图相关联的矩阵族。换言之，本文揭示了矩阵集合存在Schur形式的具体条件及其求解方法。