A Note on Kronecker Matrix Products and Matrix Equation Systems

Abstract

where B' is the transpose of B. It has been shown that Definition 1 and Theorem 1 can fruitfully be applied to problems of matrix differentiation [2]. In this note it will be shown that they can be applied to a more general class of linear matrix equations, including linear matrix differential equations. Firstly, four standard properties of Kronecker products have to be related, all of which may be proved in an elementary fashion [1, p. 223 if.]. The matrices involved can have any appropriate orders. In Property 4 it is assumed that A and B are square of order m and s, respectively. (The same order assumption will be made in Theorems 2 and 3, which will be presented further on.) PROPERTY 1. (A B)(C D) =(AC) (BD). PROPERTY 2. (A 0 B)' = A' 0 B'. PROPERTY. 3. (A + B) 09 (C + D) = A C + A D + B (& C + B X D. PROPERTY 4. If A has characteristic roots oci, i = 1, -*, m, and if B has characteristic roots /IB, j = 1,... , s, then A 0 B has characteristic roots oci,Bj. Further, Is 0 A + B 0) Im has characteristic roots oci + f3B. For the treatment of differential equations we need the matrix exponential