Two Matrix Theorems Arising from Nilpotent Groups

For a nilpotent group G without pi-torsion, and x, y is an element of G , if x(n) = y(n) for a pi-number n, then x=y ; if x(m) y(n) = y(n) x(m) for pi-numbers m, n, then xy = yx . This is a well-known 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 (ii) 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( lambda) with X = f (A). In addition, let A and B be complex nxn 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 (ii) AB = BA when A(s) B-t = B-t A(s) for two positive integers s and t.