矩阵多项式秩的和的恒等式及其应用

An Identity on the Sum of Rank of Matrix Polynomial and Its Application

证明了矩阵A的两个多项式秩的和等于它们最大公因式与最小公倍式秩的和,这个结果不仅可以概括近期文献的相关工作,而且可以对应用矩阵多项式求逆矩阵的方法作进一步的研究,同时也可使关于矩阵秩恒等式的最新讨论获得一种简单统一的处理方法.

We have proved that the sum of ranks of two polynomials of the matrix A is equal to the sum on ranks of the greatest common divisor and the smallest common multiple of matrices. It not only induces the corresponding results in the present litertures but also make a further research on the solution to the inverse of matrix polynomial. Meanwhile, it also give a uniform and simply method deal with rank identity of matrix.