Distance to nearest skew-symmetric matrix polynomials of bounded rank

TL;DR

This paper introduces an algorithm to approximate a matrix polynomial by a skew-symmetric polynomial of specified even rank, leveraging recent theoretical advances, with applications demonstrated through numerical experiments.

Contribution

The paper develops a new algorithm for approximating matrix polynomials by skew-symmetric ones of bounded rank, based on recent eigenstructure and factorization theories.

Findings

Algorithm effectively approximates matrix polynomials with skew-symmetric ones of prescribed rank.

Adapted algorithm improves performance for matrix pencils.

Numerical experiments validate the algorithm's effectiveness and compare it to existing methods.

Abstract

We propose an algorithm that approximates a given matrix polynomial of degree $d$ by another skew-symmetric matrix polynomial of a specified rank and degree at most $d$. The algorithm is built on recent advances in the theory of generic eigenstructures and factorizations for skew-symmetric matrix polynomials of bounded rank and degree. Taking into account that the rank of a skew-symmetric matrix polynomial is even, the algorithm works for any prescribed even rank greater than or equal to $2$ and produces a skew-symmetric matrix polynomial of that exact rank. We also adapt the algorithm for matrix pencils to achieve a better performance. Lastly, we present numerical experiments for testing our algorithms and for comparison to the previously known ones.