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 $U{V^{ - 1}} = \exp K$, then for every unitary-invariant norm the distance between the eigenvalues of $U$ and those of $V$ is bounded by $||K||$. This generalises two earlier results which used particular unitary-invariant norms.