A stronger form of Yamamoto's theorem on singular values

Abstract

For a matrix $T \in M_m(\mathbb{C})$, let $|T| : = \sqrt{T^*T}$. For $A \in M_m(\mathbb{C})$, we show that the matrix sequence $\big\{ |A^n|^{\frac{1}{n}} \big\}_{n \in \mathbb{N}}$ converges in norm to a positive-semidefinite matrix $H$ whose $j^{\textrm{th}}$-largest eigenvalue is equal to the $j^{\textrm{th}}$-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 $j^{\textrm{th}}$-largest eigenvalue-modulus of $A$, where $s_j(A^n)$ denotes the $j^{\textrm{th}}$-largest singular value of $A^n$. Moreover, we also discuss applications to the asymptotic behaviour of the matrix exponential function, $t \mapsto e^{tA}$.