The number of distinct eigenvalues of a matrix after perturbation

Journal article

We prove a new theorem relating the number of distinct eigenvalues of a matrix after perturbation to the prior number of distinct eigenvalues, the rank of the update, and the degree of nondiagonalizability of the matrix. In particular, a rank one update applied to a diagonalizable matrix can at most double the number of distinct eigenvalues. The theorem applies to both symmetric and nonsymmetric matrices and perturbations, of arbitrary magnitudes. An an application, we prove that in exact arithmetic the number of Krylov iterations required to exactly solve a linear system involving a diagonalizable matrix can at most double after a rank one update.

Authors: Farrell, P

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: Society for Industrial and Applied Mathematics
• Journal: SIAM Journal on Matrix Analysis and Applications
• Volume: 37
• Issue: 2
• Pages: 572-576
• Publication date: 2016-04-26
• Acceptance date: 2016-03-03
• ISSN: 0895-4798
• Keywords: perturbation; distinct eigenvalues; deflation; Krylov methods