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 same order and all coefficients are on the left (similarly for a right-unilateral equation). Recently certain perturbative solutions of unilateral equations and their properties have been discussed. We present a unified approach based on the generalized Bezout theorem for matrix polynomials. Two equations discussed in the literature, their perturbative solutions and the relation between them are described. More abstractly, the coefficients and the unknown can be taken as elements of an associative, but possibly noncommutative, algebra.