λ-矩阵的等价和矩阵多项式秩的恒等式

The Equivalence of Lamda Matrices and Rank Identities of Matrix Polynomials

设U(λ)与V(λ)都是m×m阶的λ-矩阵.若U(λ)与V(λ)等价,则对于任意的n阶方阵A,分块矩阵U(A)与V(A)的秩相等.利用此结论刻画了幂零矩阵、零化多项式等.同时,通过考虑两个对角λ-矩阵等价的充要条件,使关于矩阵多项式秩的一些恒等式的讨论有了新的统一的方法.

Let U（λ） and V（λ） be m times m lamda-matrices. If U（λ） and V（λ） are equivalent, then the rank of the block matrix U（λ） is equal to the rank of the block matrix V（λ） for any n times n matrix A. We describe the nilpotent matrices, zeroized polynomial and so on by using the conclusion as above. Meanwhile, we give a new and uniform method when dealing with the recent conclusions about the rank identities of matrix polynomials by considering the necessary and sufficient condition of equivalence between the two diagonal lamda-matrices.