Ritz values of normal matrices and Ceva’s theorem

This paper investigates the behavior of Ritz values of normal matrices. We apply Ceva’s theorem, a classical geometric result, to understand the geometric relationship between pairs of Ritz values for 3×3 normal non-Hermitian matrices, and then analyze the implications for larger matrices. We find that, in the case of normal non-Hermitian matrices, the geometric constraints on the placement of Ritz values provide less freedom than the Cauchy interlacing theorem in the Hermitian case. Using our results we analyze the restarted Arnoldi method with exact shifts applied to a 3×3 normal, non-Hermitian matrix.