Ela on a Strong Form of a Conjecture of Boyle and Handelman

On a strong form of a conjecture of Boyle and Handelman

An Upper Bound on the Characteristic Polynomial of a Nonnegative Matrix Leading to a Proof of the Boyle–handelman Conjecture

In their celebrated 1991 paper on the inverse eigenvalue problem for nonnegative matrices, Boyle and Handelman conjectured that if A is an (n+1)×(n+1) nonnegative matrix whose nonzero eigenvalues are: λ0 ≥ |λi|, i = 1, . . . , r, r ≤ n, then for all x ≥ λ0, (∗) r ∏ i=0 (x− λi) ≤ x − λ 0 . To date the status of this conjecture is that Ambikkumar and Drury (1997) showed that the conjecture is tru...

Ela on a Classic Example in the Nonnegative Inverse Eigenvalue Problem

This paper presents a construction of nonnegative matrices with nonzero spectrum τ = (3 + t, 3− t,−2,−2,−2) for t > 0. The result presented gives a constructive proof of a result of Boyle and Handelman in this special case. This example exhibits a surprisingly fast convergence of the spectral gap of τ to zero as a function of the number of zeros that are added to the spectrum.

Algebraic Shift Equivalence and Primitive Matrices

Motivated by symbolic dynamics, we study the problem, given a unital subring 5 of the reals, when is a matrix A algebraically shift equivalent over S to a primitive matrix? We conjecture that simple necessary conditions on the nonzero spectrum of A are sufficient, and establish the conjecture in many cases. If S is the integers, we give some lower bounds on sizes of realizing primitive matrices...

On the Closed-Form Solution of a Nonlinear Difference Equation and Another Proof to Sroysang’s Conjecture

The purpose of this paper is twofold. First we derive theoretically, using appropriate transformation on x(n), the closed-form solution of the nonlinear difference equation x(n+1) = 1/(±1 + x(n)), n ∈ N_0. The form of solution of this equation, however, was first obtained in [10] but through induction principle. Then, with the solution of the above equation at hand, we prove a case ...