Pseudospectra of Isospectrally Reduced Matrices and Systems

TL;DR

This paper extends the concept of pseudospectra to isospectral reductions of matrices with rational entries, showing that reduction improves eigenvalue robustness and introducing inverse pseudospectra for stability analysis.

Contribution

It generalizes isospectral reduction to matrices with rational functions, extends pseudospectrum concepts, and introduces inverse pseudospectra for stability insights.

Findings

Pseudospectrum shrinks with matrix reduction, indicating increased eigenvalue robustness.

Eigenvalues of reduced matrices are more stable under perturbations.

Inverse pseudospectra reveal stability of poles in rational function matrices.

Abstract

The isospectral reduction of matrix, which is closely related to its Schur complement, allows to reduce the size of a matrix while maintaining its eigenvalues up to a known set. Here we generalize this procedure by increasing the number of possible ways a matrix can be isospectrally reduced. The reduced matrix has rational functions as entries. We show that the notion of pseudospectrum can be extended to this class of matrices and that the pseudospectrum of a matrix shrinks as the matrix is reduced. Hence the eigenvalues of a reduced matrix are more robust to entry-wise perturbations than the eigenvalues of the original matrix. We also introduce the notion of inverse pseudospectrum (or pseudoresonances), which indicates how stable the poles of a matrix with rational function entries are to certain matrix perturbations. A mass spring system is used to illustrate and give a physical interpretation to both pseudospectra and inverse pseudospectra.