Convergence of iterated Aluthge transform sequence for diagonalizable matrices

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Given an r× r complex matrix T, if T=U|T| is the polar decomposition of T, then, the Aluthge transform is defined by Δ(T)= |T|¹/² U |T |¹/². Let Δⁿ(T) denote the n-times iterated Aluthge transform of T, i.e. Δ⁰(T)=T and Δⁿ(T)=Δ(Δⁿ⁻¹(T)), n. We prove that the sequence {Δⁿ(T)}n converges for every r× r diagonalizable matrix T. We show that the limit Δ∞(·) is a map of class C∞ on the similarity orbit of a diagonalizable matrix, and %of class C∞ on the (open and dense) set of r× r matrices with r different eigenvalues.