On M-th Roots Of Matrices Over A Field Of Positive Characterisitic

Let A be a square matrix in F,m≥ 2 is a positive integer,and X be an m-th root of A if the matrix satisfies Xm=A.Matrix equation Xm=A has played an important role in the development of matrix theory.This paper focus on the problem of when an m-th root X of A over an algebraically closed field F of positive characteristic can be expressed as a polynomial of A.Let CharF=p>0,A be an n-order matrix on F,Xbe an m-th root of A,m≥ 2.The following theorem is proved in this paper.Theorem 1(ⅰ)If p|m,X can be expressed as a polynomial in A if and only if rankX2=rankX and |σ(A)|=|σ(X)|;(ⅱ)If p|m,X can be expressed as a polynomial in A if and only if both X and A are diagonalizble and |σ(A)|=|σ(X)|.Theorem 2 If X and Y are two m-th roots of A which can be expressed as polynomials in A,then X=Y if and only if σ(X)=6(Y).In the two theorems above.Let |σ(A)| denote the set of all different eigenvalues of A,|σ(X)|denotes the set of all different eigenvalues of X.This work is a necessary complement to that of Liu Heguo-Zhao jing,who studied the same problem for the m-th roots of matrices over algebraically closed field of eigenzero in《On m-th roots of complex matrices)).