Generalized Eigenvalue Problems with Prespecified Eigenvalues

Details Generalized Eigenvalue Problems with Prespecified Eigenvalues Generalized Eigenvalue Problems with Prespecified Eigenvalues

2010-09-12

arxiv.org/abs/1009.2222v1

Mathematics

Numerical Analysis

24 pages with 2 pdf figures

Scientific paper

We consider the distance from a linear pencil A-lambda B (square or non-square) to the nearest pencil in 2-norm that has the prespecified eigenvalues lambda1, ..., lambdal with algebraic multiplicities summing up to r or greater. A singular value optimization characterization is derived for this problem under mild linear independence and multiplicity assumptions. The corollaries of the singular value optimization characterization are significant. First this provides a singular value formula to determine the nearest pencil whose eigenvalues lie in a compact region in the complex plane. Secondly this partially solves the problem posed by Boutry, Elad, Golub and Milanfar (in their SIMAX paper, vol 27 pages 582-600, published in 2005) regarding the distance from a non-square nxm pencil with n

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