Properties of perturbative solutions of unilateral matrix equations

A left-unilateral matrix equation is an algebraic equation of the form a(0)+a(1)x+a(2)x(2)+...+a(n)x(n)=0 where the coefficients a(r) and the unknown x are square matrices of the sa me order and all coefficients are on the left (similarly for a right-unilat eral equation). Recently certain perturbative solutions of unilateral equat ions and their properties have been discussed. We present a unified approac h based on the generalized Bezout theorem for matrix polynomials. Two equat ions discussed in the literature, their perturbative solutions and the rela tion between them are described. More abstractly, the coefficients and the unknown can be taken as elements of an associative, but possibly noncommuta tive, algebra.