关于矩阵多项式秩的几个恒等式

Several Identical Equations about the Ranks of the Matrix Polynomials

矩阵运算的秩一般以不等式的形式出现,给矩阵秩的计算和应用造成诸多不便.利用互素多项式乘积秩的恒等式以及方阵幂秩的分块矩阵表示,给出了一般矩阵多项式秩的分块矩阵表示以及在矩阵可以对角化情况下的一个恒等式.

The ranks of matrix operations usually show the forms of inequalities,which are inconvenient for calculation and application of the matrix ranks.However,the identical equations of the ranks of coprime polynomials,as well as the block matrix denotations of the square matrix＇s power ranks,are both concerned in this paper.Then the expressions of the ranks of general matrix polynomials are obtained by virtue of block matrices.Meanwhile,an identical equation is also received under the conditions of diagonalizable matrices.