Algorithms for Simultaneous Block Triangularization and Block Diagonalization of Sets of Matrices

TL;DR

This paper introduces new algorithms for simultaneously transforming sets of matrices into block triangular and block diagonal forms, enhancing the ability to analyze their structure and invariant subspaces.

Contribution

It presents novel algorithms for simultaneous block triangularization and diagonalization, including a new characterization and an approach based on generalized eigenvectors.

Findings

Algorithms successfully applied to concrete examples

New characterization for block diagonalization provided

Enhanced methods for analyzing matrix sets' structure

Abstract

In a recent paper, a new method was proposed to find the common invariant subspaces of a set of matrices. This paper invstigates the more general problem of putting a set of matrices into block triangular or block-diagonal form simultaneously. Based on common invariant subspaces, two algorithms for simultaneous block triangularization and block diagonalization of sets of matrices are presented. As an alternate approach for simultaneous block diagonalization of sets of matrices by an invertible matrix, a new algorithm is developed based on the generalized eigen vectors of a commuting matrix. Moreover, a new characterization for the simultaneous block diagonalization by an invertible matrix is provided. The algorithms are applied to concrete examples using the symbolic manipulation system Maple.