On the unitary 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 static matrices $A \in \mathbb {C}_{n,n}$ alike, this paper considers ways to decompose matrix flows and single matrices globally via one constant matrix similarity Cn, n as A(t) = C− 1 ⋅ diag(A1(t),...,Al(t)) ⋅ C or A = C− 1 ⋅diag(A1,...,Al) ⋅ C with each diagonal block Ak(t) or Ak square and their number l exceeding 1 if this is possible. The theory behind our proposed algorithm is elementary and uses the concept of invariant subspaces for the MATLAB eig computed ‘eigenvectors’ of one associated flow matrix B(ta) to find the coarsest simultaneous block structure for all flow matrices B(tb). The method works efficiently for all time-varying matrix flows A(t), be they real or complex, normal, with Jordan structures or repeated eigenvalues, differentiable, continuous, or discontinuous in t, and likewise for all fixed entry matrices A. Our intended aim is to discover unitarily diagonal-block decomposable flows as they originate in real-time from sensor given data for time-varying matrix problems that are unitarily invariant. Then, the complexities of their numerical treatments decrease by adopting ‘divide and conquer’ methods for their diagonal blocks. 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.