Aluthge Transforms and the Convex Hull of the Eigenvalues of a Matrix

Abstract

The Aluthge transform of a matrix is observed from the standpoint of smoothing process toward normality. We prove that every limiting point (matrix) of iterated Aluthge transforms is a normal matrix, and consequently the intersection of the numerical ranges of all iterated Aluthge transforms coincides with the convex hull of the eigenvalues of T. We prove further that the numerical range of T coincides with the convex hull of its eigenvalues if and only if T and Δ (T) have the same numerical range.

Keywords:

• Aluthge transform
• Numerical range
• Convex hull of eivenvalues
• Convexoid matrix

Acknowledgements

The author thanks Prof. T. Yamazaki and Prof. P.Y. Wu for valuable comments on the original version of the present article.