Two Matrix Theorems Arising from Nilpotent Groups

For a nilpotent group Formula: see text without Formula: see text-torsion, and Formula: see text, Formula: see text, if Formula: see text for a Formula: see text-number Formula: see text, then Formula: see text; if Formula: see text for Formula: see text-numbers Formula: see text, Formula: see text, then Formula: see text. 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 Formula: see text be a given positive integer and Formula: see text a complex square matrix satisfying that (i) all eigenvalues of Formula: see text are nonnegative, and (ii) Formula: see text; then Formula: see text has a unique Formula: see text-th root Formula: see text with Formula: see text, all eigenvalues of Formula: see text are nonnegative, and moreover there is a polynomial Formula: see text with Formula: see text. In addition, let Formula: see text and Formula: see text be complex Formula: see text matrices with all eigenvalues nonnegative, and Formula: see text, Formula: see text; then (i) Formula: see text when Formula: see text for some positive integer Formula: see text, and (ii) Formula: see text when Formula: see text for two positive integers Formula: see text and Formula: see text.