On the Decomposability of 1-Parameter Matrix Flows
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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· diag(A_1(t), ..., A_ℓ(t)) · C with each diagonal blockA_k(t) square and the number of blocks ℓ > 1 if possible. The theory behind our algorithm is elementary and uses the concept of invariant subspaces for the Matlab 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.
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