Strong majorization for Hermitian matrices
This paper concerns a partial characterization of \(G<_aF\) \((F\) is the real part of some matrix similar to \(G)\) and \(\widetilde G<_m \widetilde F\) \((\widetilde F\) is the modulus of some matrix similar to \(\widetilde G\); \(\widetilde F\), \(\widetilde G\) positive definite). The latter implies that the spectrum \(\sigma(\widetilde F)\) multiplicatively majorizes \(\sigma (\widetilde G)\). The converse is not true. Let \(G\) be real diagonal with all diagonal entries different, and let \(F\) be arbitrary Hermitian. Then \(\sigma (F)\) majorizes \(\sigma (G)\) if and only if \(G<_a F\). For diagonalizable matrices these conditions are equivalent to known multiplicative and additive spectral majorization conditions.
scientific article; zbMATH DE number 999977