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].