Analysis of Spectral Variation and Some Inequalities

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 , are unitary matrices and is a skew-Hermitian matrix such that <!-- MATH $U{V^{ - 1}} = \exp K$ --> , then for every unitary-invariant norm the distance between the eigenvalues of and those of is bounded by . This generalises two earlier results which used particular unitary-invariant norms.