On the Block-Decomposability of 1-Parameter Matrix Flows and Static Matrices

TL;DR

This paper presents an elementary, efficient method for globally decomposing 1-parameter matrix flows and static matrices into block-diagonal forms using invariant subspaces, applicable to various matrix types and structures.

Contribution

It introduces a new algorithm leveraging eigenvector invariants to find the coarsest block structure for matrix flows and static matrices, enhancing numerical treatment efficiency.

Findings

Method works for differentiable, continuous, or discontinuous flows.

Applicable to hermitean, symmetric, normal, and general matrices.

Discovers and studies k-normal matrix classes for block decomposition.

Abstract

For general complex or real 1-parameter matrix flows $A(t)_{n,n}$ and for time-invariant static matrices $A \in \CC_{n,n}$ alike, this paper considers ways to decompose matrix flows and single matrices globally via one constant matrix similarity $C_{n,n}$ as $A(t) = C ^{-1} \cdot \text{ diag}(A_1(t), ..., A_\ell(t)) \cdot C$ or $A = C^{-1}\cdot \text{diag}(A_1,...,A_\ell)\cdot C$ with each diagonal block $A_k(t)$ or $A_k$ square and their number $\ell > 1$ if this is possible. The theory behind our proposed algorithm is elementary and uses the concept of invariant subspaces for the Matlab {\tt eig} computed 'eigenvectors' of one associated flow matrix $B(t_a)$ to find the coarsest simultaneous block structure for all flow matrices $B(t_b)$. The method works very efficiently for all time-varying matrix flows, be they differentiable, continuous or discontinuous in $t$, and for all fixed entry matrices $A$; as well as for all types of square matrix flows or fixed entry matrices such as hermitean, real symmetric, normal or general complex and real flows $A(t)$ or static matrices $A$, with or without Jordan block structures and with or without repeated eigenvalues. Our intended aim is to discover diagonal-block decomposable flows as they originate in sensor driven outputs for time-varying matrix problems and thereby help to reduce the complexities of their numerical treatments through adapting 'divide and conquer' methods for their diagonal sub-blocks. Our method is also applicable to standard fixed entry matrices of all structures and types. In the process we discover and study k-normal fixed entry matrix classes that can be decomposed under unitary similarities into various $k$-dimensional block-diagonal forms.