  Given an $r\times r$ complex matrix $T$, if $T=U|T|$ is the polar
decomposition of $T$, then, the Aluthge transform is defined by $$ \Delta(T)=
|T|^{1/2} U |T |^{1/2}. $$ Let $\Delta^{n}(T)$ denote the n-times iterated
Aluthge transform of $T$, i.e. $\Delta^{0}(T)=T$ and
$\Delta^{n}(T)=\Delta(\Delta^{n-1}(T))$, $n\in\mathbb{N}$. We prove that the
sequence $\{\Delta^{n}(T)\}_{n\in\mathbb{N}}$ converges for every $r\times r$
{\bf diagonalizable} matrix $T$. We show that the limit
$\Delta^{\infty}(\cdot)$ is a map of class $C^\infty$ on the similarity orbit
of a diagonalizable matrix, and %of class $C^\infty$ on the (open and dense)
set of $r\times r$ matrices with $r$ different eigenvalues.
