An extension of Yamamoto's theorem on the eigenvalues and singular values of a matrix

Abstract

We extend, in the context of real semisimple Lie group, a result of T. Yamamoto which asserts that ${\lim }_{m\to \mathrm{\infty }}\left[s_i\right(X^m){]}^{1/m}=|{\lambda }_i\left(X\right)|$, $i=1,\dots ,n$, where $s_1\left(X\right)\ge \cdots \ge s_n\left(X\right)$ are the singular values, and ${\lambda }_1\left(X\right),\dots ,{\lambda }_n\left(X\right)$ are the eigenvalues of the $n\times n$ matrix $X$, in which $\left|{\lambda }_1\right(X\left)\right|\ge \cdots \ge \left|{\lambda }_n\right(X\left)\right|$.