For general complex or real 1-parameter matrix flow A(t)_{n,n} \CC this paper considers ways to decompose flows globally via one constant matrix  C_{n,n} as A(t) = C ^{-1} \cdot diag(A_1(t), ..., A_\ell(t)) \cdot C \cdot diag(A_1,...,A_\ell)\cdot C } with each diagonal blockA _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 .
For general complex or real 1-parameter matrix flows A(t)_{n,n} and for time-invariant static matrices A \in\CC_{n,n 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 diag(A_1(t), ..., A_\ell(t)) \cdot C or A = C^{-1\cdot 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 .
R
flow
flows
A
and for time-invariant static matrices A \in
A
_{n,n
R
flows
matrix flows and single matrices
A
similarity
A
or A = C^{-1
R
blockA
block A
R
square and the number of blocks
or A_k square and their number
A
this is
A
proposed
R
flow matrix A
associated flow matrix B
R
A
B
A
time-varying
A
fixed entry matrices A; as well as for all
A
or fixed entry matrices
A
or static matrices A
A
diagonal-block
R
given
driven
A
help to
R
treatment
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