The iterated Aluthge transforms of a matrix converge

Given an r×r complex matrix T, if T=U|T| is the polar decomposition of T, then, the Aluthge transform is defined byΔ(T)=|T|1/2U|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∈N. We prove that the sequence {Δn(T)}n∈N converges for every r×r matrix T. This result was conjectured by Jung, Ko and Pearcy in 2003. We also analyze the regularity of the limit function.