A stronger form of Yamamoto's theorem on singular values

Abstract

For a matrix $T\in M_m(\mathbb{C})$, let $|T|:=\sqrt{T^{\ast }T}$. For $A\in M_m(\mathbb{C})$, we show that the matrix sequence $\{|A^n{|}^{\frac{1}{n}}{\}}_{n\in \mathbb{N}}$ converges in norm to a positive-semidefinite matrix $H$ whose $j^{\text{th}}$-largest eigenvalue is equal to the $j^{\text{th}}$-largest eigenvalue-modulus of $A$ (for $1\le j\le m$). In fact, we give an explicit description of the spectral projections of $H$ in terms of the eigenspaces of the diagonalizable part of $A$ in its Jordan-Chevalley decomposition. This gives us a stronger form of Yamamoto's theorem which asserts that ${\lim }_{n\to \infty }{s}_j(A^n{)}^{\frac{1}{n}}$ is equal to the $j^{\text{th}}$-largest eigenvalue-modulus of $A$, where $s_j(A^n)$ denotes the $j^{\text{th}}$-largest singular value of $A^n$. Moreover, we also discuss applications to the asymptotic behaviour of the matrix exponential function, $t\mapsto e^{tA}$.

13 pages, minor changes. Accepted for publication in LAA