  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, $t \mapsto e^{tA}$.
