On the Inverse of a Square Polynomial Matrix

Abstract

In this work, we explore two methods for finding the inverse of polynomial matrices: the Gauss-Jordan inversion method and the Yujiro Inouye algorithm. The Gauss-Jordan method applies to the inversion of polynomial matrices and necessitates operations involving polynomials. Notably, when performing these operations, the resultant inverse may contain polynomials of high degree if common factors in the divisor and dividend polynomials are not canceled out in the numerators and denominators. Conversely, the Yujiro Inouye algorithm requires only operations with constant matrices. This algorithm produces an inverse in minimal degree form, provided that the polynomial matrix being inverted is not of a special form. It has been demonstrated that this method is faster than existing alternatives. Several examples are provided to illustrate the feasibility of both methods.