Analysis of spectral variation and some inequalities

Abstract

A geometric method, based on a decomposition of the space of complex matrices, is employed to study the variation of the spectrum of a matrix. When adapted to special cases, this leads to some classical inequalities as well as some new ones. As an example of the latter, we show that if U, V are unitary matrices and K is a skew-Hermitian matrix such that UV^-1 = exp K, then for every unitary-invariant norm the distance between the eigenvalues of V and those of Vis bounded by ||K||.This generalises two earlier results which used particular unitary-invariant norms.