An Algebraic Approach to Non-Orthogonal General Joint Block Diagonalization

Abstract

The exact/approximate non-orthogonal general joint block diagonalization ({\sc nogjbd}) problem of a given real matrix set $\mathcal{A}=\{A_i{\}}_{i=1}^m$ is to find a nonsingular matrix $W\in {\mathbb{R}}^{n\times n}$ (diagonalizer) such that $W^T{A}_{i}W$ for $i=1,2,\dots ,m$ are all exactly/approximately block diagonal matrices with the same diagonal block structure and with as many diagonal blocks as possible. In this paper, we show that a solution to the exact/approximate {\sc nogjbd} problem can be obtained by finding the exact/approximate solutions to the system of linear equations $A_{i}Z=Z^T{A}_i$ for $i=1,\dots ,m$, followed by a block diagonalization of $Z$ via similarity transformation. A necessary and sufficient condition for the equivalence of the solutions to the exact {\sc nogjbd} problem is established. Two numerical methods are proposed to solve the {\sc nogjbd} problem, and numerical examples are presented to show the merits of the proposed methods.