摘要
For a nilpotent group G without π-torsion,and x,y ∈ G,if x^(n)=y^(n) for a T-number n,then x=y;if x^(m)y^(n)=y^(n)x^(m) for n-numbers m,n,then xy=yx.This is a wellknown result in group theory.In this paper,we prove two analogous theorems on matrices,which have independence significance.Specifically,let m be a given positive integer and A a complex square matrix satisfying that(i)all eigenvalues of A are nonnegative,and(i)rank A^(2)=rank A;then A has a unique m-th root X with rank X^(2)=rank X,all eigenvalues of X are nonnegative,and moreover there is a polynomial f(λ)with X=f(A).In addition,let A and B be complex n×n matrices with all eigenvalues nonnegative,and rank A^(2)=rank A,rank B^(2)=rank B;then(i)A=B when A^(r)=B^(r) for some positive integer r,and(i)AB=BA when A^(s)B^(t)=B^(t)A^(s) for two positive integers s and t.
基金
Supported by National Natural Science Foundation of China(No.12171142).
