arXiv subjects

Benjamin J. Clark

Publications and source records attributed to Benjamin J. Clark.

5 recordsLinked to original sources

arXiv2024

Properties of the cone of polynomials of fixed degree that preserve nonnegative matrices

As was detailed by Loewy and London in [Linear and Multilinear Algebra 6 (1978/79), no.~1, 83--90], the cone of polynomials that preserve the nonnegativity of matrices may play an important role in the solution to the nonnegative inverse eigenvalue problem. In this paper, we start by showing the cone generated by polynomials of degree greater than or equal to $2n$ that preserve nonnegative matrices of order $n$ is non-polyhedral. Next, a question posed by Loewy in [Linear Algebra and its Applications, 676(2023), 267--276], about how negative the center term can be in a degree $2n$ polynomial is answered. We extend this to show that a polynomial that preserves nonnegative matrices of order $n$ can have it's the largest term, in absolute value, be arbitrarily negative with the remaining coefficients being one. We conclude, by exploring properties of the measure of the cone when restricted to the unit sphere and by proving some initial bounds of that volume.

math.RA↗

arXiv2024

The NIEP is solvable by reality and finitely many polynomial inequalities

The nonnegative inverse eigenvalue problem (NIEP) is shown to be solvable by the reality condition, spectrum equal to its conjugate, as well as by a finite union and intersection of polynomial inequalities. It is also shown that the symmetric NIEP and real NIEP form semi-algebraic sets and can therefore be solved just by a finite union and intersection of polynomial inequalities. An overview of ideas are given in how tools from real algebraic geometry may be applied to the NIEP and related sub-problems.

math.AG↗

arXiv2024

Polynomials that preserve nonnegative monomial matrices

A recently-established necessary condition for polynomials that preserve the class of entrywise nonnegative matrices of a fixed order is shown to be necessary and sufficient for the class of nonnegative monomial matrices. Along the way, we provide a formula for computing an arbitrary power of a monomial matrix and a formula for computing the polynomial of a nonnegative monomial matrix.

math.RA↗

arXiv2022

Polynomials that preserve nonnegative matrices of order two

A known characterization for entire functions that preserve all nonnegative matrices of order two is shown to characterize polynomials that preserve nonnegative matrices of order two. Equivalent conditions are derived and used to prove that $\mathscr{P}_3 \subset \mathscr{P}_2$, which was previously unknown. A new characterization is given for polynomials that preserve nonnegative circulant matrices of order two.

math.RA↗

arXiv2021

Polynomials that preserve nonnegative matrices

In further pursuit of a solution to the celebrated nonnegative inverse eigenvalue problem, Loewy and London [Linear and Multilinear Algebra 6 (1978/79), no.~1, 83--90] posed the problem of characterizing all polynomials that preserve all nonnegative matrices of a fixed order. If $\mathscr{P}_n$ denotes the set of all polynomials that preserve all $n$-by-$n$ nonnegative matrices, then it is clear that polynomials with nonnegative coefficients belong to $\mathscr{P}_n$. However, it is known that $\mathscr{P}_n$ contains polynomials with negative entries. In this work, novel results for $\mathscr{P}_n$ with respect to the coefficients of the polynomials belonging to $\mathscr{P}_n$. Along the way, a generalization for the even-part and odd-part are given and shown to be equivalent to another construction that appeared in the literature. Implications for further research are discussed.

math.RA↗