Constructing New Realisable Lists from Old in the NIEP

Given a list of complex numbers \sigma:=(\lambda_1,\lambda_2,...,\lambda_m), we say that {\sigma} is realisable if {\sigma} 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 (\rho,\lambda_2,\lambda_3,...,\lambda_m), where {\rho} is the Perron eigenvalue and \lambda_2 is real, we find families of lists (\mu_1,\mu_2,...,\mu_n), for which (\mu_1,\mu_2,...,\mu_n,\lambda_3,\lambda_4,...,\lambda_m) is realisable. In addition, given a realisable list (\rho,\alpha+i\beta,\alpha-i\beta,\lambda_4,\lambda_5,...,\lambda_m), where {\rho} is the Perron eigenvalue and {\alpha} and {\beta} are real, we find families of lists (\mu_1,\mu_2,\mu_3,\mu_4), for which (\mu_1,\mu_2,\mu_3,\mu_4,\lambda_4,\lambda_5,...,\lambda_m) is realisable

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

The maximum of the minimal multiplicity of eigenvalues of symmetric matrices whose pattern is constrained by a graph

In this paper we introduce a parameter $Mm(G)$, defined as the maximum over the minimal multiplicities of eigenvalues among all symmetric matrices corresponding to a graph $G$. We compute $Mm(G)$ for several families of graphs

Families of Newton-like inequalities for sets of self-conjugate complex numbers

We derive families of Newton-like inequalities involving the elementary symmetric functions of sets of self-conjugate complex numbers in the right half-plane. These are the first known inequalities of this type which are independent of the proximity of the complex numbers to the real axis

Graphs that allow all the eigenvalue multiplicities to be even

Let $G$ be an undirected graph on $n$ vertices and let $S(G)$ be the set of all $n \times n$ real symmetric matrices whose nonzero off-diagonal entries occur in exactly the positions corresponding to the edges of $G$. The inverse eigenvalue problem for a graph $G$ is a problem of determining all possible lists that can occur as the lists of eigenvalues of matrices in $S(G).$ This question is, in general, hard to answer and several variations were studied, most notably the minimum rank problem. In this paper we introduce the problem of determining for which graphs $G$ there exists a matrix in $S(G)$ whose characteristic polynomial is a square, i.e. the multiplicities of all its eigenvalues are even. We solve this question for several families of graphs

Integer completely positive matrices of order two

We show that every integer doubly nonnegative $2 \times 2$ matrix has an integer cp-factorization

The effect of assuming the identity as a generator on the length of the matrix algebra

Let $M_n(\mathbb{F})$ be the algebra of $n \times n$ matrices and let $\mathcal S$ be a generating set of $M_n(\mathbb{F})$ as an $\mathbb{F}$-algebra. The length of a finite generating set $\mathcal S$ of $M_n(\mathbb{F})$ is the smallest number $k$ such that words of length not greater than $k$ generate $M_n(\mathbb{F})$ as a vector space. Traditionally the identity matrix is assumed to be automatically included in all generating sets $\mathcal S$ and counted as a word of length $0$. In this paper we discuss how the problem changes if this assumption is removed.Comment: 15 page

A Nordhaus-Gaddum conjecture for the minimum number of distinct eigenvalues of a graph

We propose a Nordhaus-Gaddum conjecture for $q(G)$, the minimum number of distinct eigenvalues of a symmetric matrix corresponding to a graph $G$: for every graph $G$ excluding four exceptions, we conjecture that $q(G)+q(G^c)\le |G|+2$, where $G^c$ is the complement of $G$. We compute $q(G^c)$ for all trees and all graphs $G$ with $q(G)=|G|-1$, and hence we verify the conjecture for trees, unicyclic graphs, graphs with $q(G)\le 4$, and for graphs with $|G|\le 7$

The integer cp-rank of 2×22 \times 2 matrices

We show the cp-rank of an integer doubly nonnegative $2 \times 2$ matrix does not exceed $11$

Diagonal realizability in the Nonnegative Inverse Eigenvalue Problem

We show that if a list of nonzero complex numbers $\sigma=(\lambda_1,\lambda_2,\ldots,\lambda_k)$ is the nonzero spectrum of a diagonalizable nonnegative matrix, then $\sigma$ is the nonzero spectrum of a diagonalizable nonnegative matrix of order $k+k^2.