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

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.