Before revision:  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 .
After revision:  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 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 .