Ritz Value Localization for Non-Hermitian Matrices

Rayleigh-Ritz eigenvalue estimates for Hermitian matrices obey Cauchy interlacing,which has helpful implications for theory, applications, and algorithms. In contrast, few resultsabout the Ritz values of non-Hermitian matrices are known, beyond their containment within thenumerical range. To show that such Ritz values enjoy considerable structure, we establish regionswithin the numerical range in which certain Ritz values of general matrices must be contained. Todemonstrate that localization occurs even for extreme examples, we carefully analyze possible Ritzvalue combinations for a three-dimensional Jordan block