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Richard Ellard

Publications and source records attributed to Richard Ellard.

5 recordsLinked to original sources
arXiv2017

Diagonal elements in the Nonnegative Inverse Eigenvalue Problem

We say that a list of complex numbers is "realisable" if it is the spectrum of some (entrywise) nonnegative matrix. The Nonnegative Inverse Eigenvalue Problem (NIEP) is the problem of characterising all realisable lists. Although the NIEP remains unsolved, it has been solved in the case where every entry in the list (apart from the Perron eigenvalue) has nonpositive real part. For a given spectrum of this type, we show that a list of nonnegative numbers may arise as the diagonal elements of the realising matrix if and only if these numbers satisfy a remarkably simple inequality. Furthermore, we show that realisation can be achieved by the sum of a companion matrix and a diagonal matrix.

math.SP↗
arXiv2017

An extension of the Hermite-Biehler theorem with application to polynomials with one positive root

If a real polynomial $f(x)=p(x^2)+xq(x^2)$ is Hurwitz stable (every root if $f$ lies in the open left half-plane), then the Hermite-Biehler Theorem says that the polynomials $p(-x^2)$ and $q(-x^2)$ have interlacing real roots. We extend this result to general polynomials by giving a lower bound on the number of real roots of $p(-x^2)$ and $q(-x^2)$ and showing that these real roots interlace. This bound depends on the number of roots of $f$ which lie in the left half plane. Another classical result in the theory of polynomials is Descartes' Rule of Signs, which bounds the number of positive roots of a polynomial in terms of the number of sign changes in its coefficients. We use our extension of the Hermite-Biehler Theorem to give an inverse rule of signs for polynomials with one positive root.

math.CA↗
arXiv2015

Connecting sufficient conditions for the Symmetric Nonnegative Inverse Eigenvalue Problem

We say that a list of real numbers is "symmetrically realisable" if it is the spectrum of some (entrywise) nonnegative symmetric matrix. The Symmetric Nonnegative Inverse Eigenvalue Problem (SNIEP) is the problem of characterising all symmetrically realisable lists. In this paper, we present a recursive method for constructing symmetrically realisable lists. The properties of the realisable family we obtain allow us to make several novel connections between a number of sufficient conditions developed over forty years, starting with the work of Fiedler in 1974. We show that essentially all previously known sufficient conditions are either contained in or equivalent to the family we are introducing.

math.SP↗
arXiv2013

Constructing New Realisable Lists from Old in the NIEP

Given a list of complex numbers σ:=(λ_1,λ_2,...,λ_m), we say that σ is realisable if σ is the spectrum of some (entrywise) nonnegative matrix. The Nonnegative Inverse Eigenvalue Problem (or NIEP) is the problem of categorising all realisable lists. Given a realisable list (ρ,λ_2,λ_3,...,λ_m), where ρ is the Perron eigenvalue and λ_2 is real, we find families of lists (μ_1,μ_2,...,μ_n), for which (μ_1,μ_2,...,μ_n,λ_3,λ_4,...,λ_m) is realisable. In addition, given a realisable list (ρ,α+iβ,α-iβ,λ_4,λ_5,...,λ_m), where ρ is the Perron eigenvalue and α and β are real, we find families of lists (μ_1,μ_2,μ_3,μ_4), for which (μ_1,μ_2,μ_3,μ_4,λ_4,λ_5,...,λ_m) is realisable.

math.SP↗