利用线性空间理论研究矩阵多项式的秩

A Research on the Rank of Matrix Polynomials Based on Linear Space Theory

摘要: 本文围绕矩阵多项式的秩展开讨论，从一个特殊的矩阵多项式的秩的命题出发，提炼出其代数本质，对其进行推广，利用线性空间理论，引出了两个一般的互素的多项式作用下的矩阵多项式秩的恒等关系式，以及推广至s个矩阵多项式的情形，并给出了严格的证明。

Abstract: This paper discusses the rank of matrix polynomials. We start from the rank proposition of a special matrix polynomial, extract its algebraic essence, and generalize it. Using linear space theory, the identity relationship of the rank of matrix polynomials under the action of two general coprime polynomials is derived, as well as extended to the case of s matrix polynomials, and a rigorous proof is given.