  Let $\lambda \in (0,1)$ and let $T$ be a $r\times r$ complex matrix with
polar decomposition $T=U|T|$. Then, the $\la$- Aluthge transform is defined by
$$ \Delta_\lambda (T )= |T|^{\lambda} U |T |^{1-\lambda}. $$ Let
$\Delta_\lambda^{n}(T)$ denote the n-times iterated Aluthge transform of $T$,
$n\in\mathbb{N}$. We prove that the sequence
$\{\Delta_\lambda^{n}(T)\}_{n\in\mathbb{N}}$ converges for every $r\times r$
{\bf diagonalizable} matrix $T$. We show regularity results for the two
parameter map $(\la, T) \mapsto \alulit{\infty}{T}$, and we study for which
matrices the map $(0,1)\ni \lambda \mapsto \Delta_\lambda^{\infty}(T)$ is
constant.
