A note on the solution of the unilateral matrix equation

Abstract

M. H. Ingraham [1] has developed an algorithm for the solution of the unilateral matrix equation Xi=o RiX =O, where the coefficients Ri (i=0, 1, 2, * * *, n) are nXn matrices with elements in a field J of characteristic zero. It is the purpose of this note to extend his algorithm to include the unilateral equation in which the coefficients are m Xn matrices. It is shown that the solution in this case is easily reduced to the solution of an equation with n Xn matrix coefficients. If in particular m i, i=1, 2, * * * , n), where the elements aii are so chosen that the degree of IT=,1 aii is less than or equal to j, and the degree of Hif.1 aii is equal to n. (3). The necessary and sufficient condition that A (A) = 0 A Ail be the c.t.f. of a matrix A-X is that W1=(A8, A8s1, A , A1) be of rankn [3]. Since X is of necessity square of order n, fact 3 is still valid. The first fact is based upon a factor theorem, namely, X is a solution of R(X) =0 if and only if R(A) =S(A)(A-X). The dimensions of the coefficients do not affect this factor theorem. However, steps 1 and 2 involve the concept of the c.t.f. of R(A) and must be reconsidered. The cases m > n and m n. Let