Convergence of iterated Aluthge transform sequence for diagonalizable matrices II: $\lambda$-Aluthge transform

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