Ritz Values of Normal Matrices and Ceva's Theorem
Hansen, Derek J.
The Cauchy interlacing theorem for Hermitian matrices provides an indispensable tool for understanding eigenvalue estimates and various numerical algorithms that rely on the Ritz values of a matrix. No generalization of interlacing is known for non-Hermitian matrices, and as a consequence, many useful algorithms for such matrices are not fully understood. Toward filling this gap, we consider the behavior of Ritz values of normal matrices. We apply Ceva's theorem, a classical geometric result, to understand two Ritz values of a 3x3 normal matrix and analyze the implications for larger matrices. Unlike the Hermitian case, specifying at most half of the Ritz values significantly restricts where the remaining Ritz values may fall. We use our results to analyze the restarted Arnoldi method with exact shifts applied to a 3 x 3 normal, non-Hermitian matrix.
Citable link to this pagehttps://hdl.handle.net/1911/102189
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