Spatial decomposition of functionally commutative matrices

Abstract

Let I ⊆ R and D ⊆ C , where R and C are the fields of real and complex numbers, respectively. Let C n×n be the space of square matrices of order n over C . A matrix-valued function F : I → C n×n is said to be proper on I if F ( t ) = f ( t , A ), where A ϵ C n×n and f : I × D → C is a scalar function, and F is said to be semiproper on I if F ( t ) F ( t ) = F ( τ ) F ( t ) for all t , τ ϵ I . The main results presented here are: (1) a characterization together with some potentially useful results for proper matrix functions; (2) a characterization of semiproper matrix functions in terms of proper ones; (3) a new and systematic procedure for decomposing a semiproper matrix function into a finite sum of mutually commutative proper ones. Some important applications of these new results in control engineering, linear systems theory, and the theory of linear differential equations are also included.