Block-Triangularizations of Partitioned Matrices Under Similarity/Equivalence Transformations

Abstract

A partitioned matrix, of which the column- and row-sets are divided into certain numbers of groups, arises from a mathematical formulation of discrete physical or engineering systems. This paper addresses the problem of the block-triangularization of a partitioned matrix under similarity/equivalence transformation with respect to its partitions of the column- and row-sets. Such block-triangularization affords a mathematical representation of the hierarchical decomposition of a physical system into subsystems if the transformation used is of physical significance. A module is defined from a partitioned matrix, and the simplicity of the module is proved to be equivalent to the nonexistence of a nontrivial block-triangular decomposition. Moreover, the existence and the uniqueness of the block-triangular forms are deduced from the Jordan--Holder theorem for modules. The results cover many block-triangularization methods hitherto discussed in the literature such as the Jordan normal form and the strongly connected-component decomposition in the case of partition-respecting similarity transformations, and the rank normal form, the Dulmage--Mendelsohn decomposition, and the combinatorial canonical form of layered mixed matrices in the case of partition-respecting equivalence transformations.