Majorization and the Schur-Horn Theorem.

Abstract

We study majorization in Rn and some of its properties. The concept of majorization plays an important role in matrix analysis by producing several useful relationships. We find out that there is a strong relationship between majorization and doubly stochastic matrices; this relation has been perfectly described in Birkhoff's Theorem. On the other hand, majorization characterizes the connection between the eigenvalues and the diagonal elements of self adjoint matrices. This relation is summarized in the Schur-Horn Theorem. Using this theorem, we prove versions of Kadison's Carpenter's Theorem. We discuss A. Neumann's extension of the concept of majorization to in_nite dimension to that provides a Schur-Horn Theorem in this context. Finally, we detail the work of W. Arveson and R.V. Kadison in proving a strict Schur-Horn Theorem for positive trace-class operators.