proof of Schur’s inequality

By Schur’s theorem, a unitary matrixMathworldPlanetmath U and an upper triangular matrixMathworldPlanetmath T exist such that A=UTUH, T being diagonal if and only if A is normal. Then AHA=UTHUHUTUH=UTHTUH, which means AHA and THT are similar; so they have the same trace. We have:



If and only if A is normal, T=D and therefore equality holds.

Title proof of Schur’s inequality
Canonical name ProofOfSchursInequality
Date of creation 2013-03-22 15:35:25
Last modified on 2013-03-22 15:35:25
Owner Andrea Ambrosio (7332)
Last modified by Andrea Ambrosio (7332)
Numerical id 7
Author Andrea Ambrosio (7332)
Entry type Proof
Classification msc 26D15
Classification msc 15A42