On the Decomposability of 1-Parameter Matrix Flows.
Frank Uhlig · arXiv (Cornell University) · 2020
For general complex or real 1-parameter matrix flow $A(t)_{n,n}$ this paper considers ways to decompose flows globally via one constant matrix $C_{n,n}$ as $A(t) = C ^{-1} \cdot \text{ diag}(A_1(t), ..., A_\ell(t)) \cdot C$ with each diagonal block$A_k(t)$ square and the number of blocks $\ell > 1$ if possible. The theory behind our algorithm is elementary and uses the concept of invariant subspaces for the Matlab {\tt eig} computed 'eigenvectors' of one flow matrix $A(t_a)$ to find the coarsest simultaneous block structure for all flow matrices $A(t_b)$. The method works very efficiently for all matrix flows, be they differentiable, continuous or discontinuous in $t$, and for all types of square matrix flows such as hermitean, real symmetric, normal or general complex and real flows $A(t)$, with or without Jordan block structures and with or without repeated eigenvalues. Our intended aim is to discover decomposable flows as they originate in sensor given outputs for time-varying matrix problems and thereby reduce the complexities of their numerical treatment.