Pseudospectra of isospectrally reduced 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.

Characterizing cospectral vertices via isospectral reduction

“…Equivalently, an isospectral reduction of a matrix is a way of taking a matrix and constructing a smaller matrix whose entries are rational functions, in a way that preserves the matrix' spectral properties. Isospectral reductions have been used to improve the eigenvalue approximations of Gershgorin, Brauer, and Brualdi [11,1,2]; study the pseudo-spectra of graphs and matrices [21]; create stability preserving transformations of networks [3,4,19]; and study the survival probabilities in open dynamical systems [5].…”

Eigenvectors of isospectral graph transformations

“…These transformations allow changing the topology of a network (modifying the interactions, reducing or increasing the number of nodes), while maintaining properties related to the network's dynamics. In [6,10] the authors relate the pseudospectrum of a graph to the pseudospectrum of its reduction.…”

Designing pretty good state transfer via isospectral reductions

“…We first provide some key aspects of isospectral reductions [13][14][15][16][17]19], introduced first by Bunimovich and Webb [13]. This concept will allow us to extract the polynomials P ± .…”

“…Inspecting this isospectral reduction R S (H, λ) in more detail, we see that the respective "on-site potentials" h 2 (λ−3) (λ−3)λ−1 and h 2 (λ−3) (λ−3)λ−1 of sites 1 and 2 become equal for all λ if and only if h = ±h . In this case (16) becomes bisymmetric, i.e., symmetric about both the diagonal and the anti-diagonal. Interestingly, the choice h = ±h is also the only one that makes u and v cospectral.…”

“…Namely, by directly obtaining the polynomials P ± from an underlying symmetric Hamiltonian that features cospectral sites u and v. To this end, we combine the mathematical relations underlying the works in Ref. [11,12] with the theory of isospectral reductions [13][14][15][16][17][18][19]. Isospectral reduction is a method to reduce the size of a given Hamiltonian while keeping a large amount of information on its eigenvalues and eigenvectors.…”