Nearest matrix polynomials with a specified elementary divisor

TL;DR

This paper investigates the minimal perturbations needed to modify a matrix polynomial so it has a specified elementary divisor, using optimization techniques and bounds, with applications in stability analysis.

Contribution

It introduces a novel approach to compute the distance to a matrix polynomial with a given elementary divisor, including regular and non-regular cases, using structured perturbations and optimization.

Findings

Distance can be characterized via two optimization problems.

Algorithms like BFGS and global search effectively compute the distance.

Upper and lower bounds for the distance are established.

Abstract

The problem of finding the distance from a given $n \times n$ matrix polynomial of degree $k$ to the set of matrix polynomials having the elementary divisor $(\lambda-\lambda_0)^j, \, j \geqslant r,$ for a fixed scalar $\lambda_0$ and $2 \leqslant r \leqslant kn$ is considered. It is established that polynomials that are not regular are arbitrarily close to a regular matrix polynomial with the desired elementary divisor. For regular matrix polynomials the problem is shown to be equivalent to finding minimal structure preserving perturbations such that a certain block Toeplitz matrix becomes suitably rank deficient. This is then used to characterize the distance via two different optimizations. The first one shows that if $\lambda_0$ is not already an eigenvalue of the matrix polynomial, then the problem is equivalent to computing a generalized notion of a structured singular value. The distance is computed via algorithms like BFGS and Matlab's globalsearch algorithm from the second optimization. Upper and lower bounds of the distance are also derived and numerical experiments are performed to compare them with the computed values of the distance.