Matrix Polynomials with Specified Eigenvalues

TL;DR

This paper develops methods to compute the minimal perturbation needed to modify a matrix polynomial so that it has a specified set of eigenvalues, using singular value optimization techniques.

Contribution

It introduces singular value optimization formulas for the distance to polynomials with prescribed eigenvalues, enabling practical computation.

Findings

Derived explicit formulas for the distance using singular values.

Provided numerical methods exploiting Lipschitzness and analyticity.

Demonstrated applicability for small sets of eigenvalues.

Abstract

This work concerns the distance in 2-norm from a matrix polynomial to a nearest polynomial with a specified number of its eigenvalues at specified locations in the complex plane. Perturbations are allowed only on the constant coefficient matrix. Singular value optimization formulas are derived for these distances facilitating their computation. The singular value optimization problems, when the number of specified eigenvalues is small, can be solved numerically by exploiting the Lipschitzness and piece-wise analyticity of the singular values with respect to the parameters.