Schur decomposition of several matrices

TL;DR

This paper characterizes when collections of matrices can be simultaneously reduced to Schur form using unitary or orthogonal transformations, linking the problem to pseudoforest graph structures.

Contribution

It provides a complete characterization of collections of matrices that can be simultaneously reduced to Schur form, based on their associated graph structures.

Findings

Characterization of matrices reducible to Schur form via pseudoforest graphs

Conditions for simultaneous Schur reduction of matrix collections

Method to find Schur form when reducible

Abstract

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.