We study the bisymmetric nonnegative inverse eigenvalue problem (BNIEP). This problem is the problem of finding the necessary and sufficient conditions on a list of $n$ complex numbers to be a spectrum of an $n \times n$ bisymmetric nonnegative matrix. Most recently, some of the sufficient conditions for the BNIEP are given by Julio and Soto in 2015. In this article, we give another proof of one result (Theorem 4.3) in [Julio and Soto, 2015] and we obtain the result very similar to the one (Theorem 4.2) in [Julio and Soto, 2015] using a different method to construct our desired bisymmetric nonnegative matrix. We also give some sufficient conditions for the BNIEP based on the sufficient conditions for the nonnegative inverse eigenvalue problem (NIEP) given by Borobia in 1995. We give the condition that is both necessary and sufficient for the BNIEP when $n \leq 4$ and then we show that for $n = 6$, the BNIEP and the symmetric nonnegative eigenvalue problem (SNIEP) are different. Moreover, some sufficient conditions for the bisymmetric positive inverse eigenvalue problem are provided. Finally, we give a new result on a sufficient condition for the BNIEP with the prescribed diagonal entries.
We give sufficient conditions of the nonnegative inverse eigenvalue problem (NIEP) for normal centrosymmetric matrices. These sufficient conditions are analogous to the sufficient conditions of the NIEP for normal matrices given by Xu [16] and Julio, Manzaneda and Soto [2].
We show that if a list of nonzero complex numbers $σ=(λ_1,λ_2,\ldots,λ_k)$ is the nonzero spectrum of a diagonalizable nonnegative matrix, then $σ$ is the nonzero spectrum of a diagonalizable nonnegative matrix of order $k+k^2.$
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, w…
We first prove two new spectral properties for symmetric nonnegative tensors. We prove a maximum property for the largest H-eigenvalue of a symmetric nonnegative tensor, and establish some bounds for this eigenvalue via row sums of that tensor. We show that if an eigenvalue of a symmetric nonnegative tensor has a positive H-eigenvector, then this eigenvalue is the largest H-eigenvalue of that tensor. We also give a necessary and sufficient condition for this. We then introduce copositive tensors. This concept extends the concept of copositive matrices. Symmetric nonnegative tensors and positiv…
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 prev…
In this paper we prove the optimal upper bound $\frac{λ_{n}}{λ_{m}}\leq\frac{n^{2}}{m^{2}}$ $\Big(λ_{n}>λ_{m}\geq 11\sup\limits_{x\in[0,1]}q(x)\Big)$ for one-dimensional Schrodinger operators with a nonnegative differentiable and single-barrier potential $q(x)$, such that $\mid q'(x) \mid\leq q^{*},$ where $q^{*}=\frac{2}{15}\min\{q(0) , q(1)\}$. In particular, if $q(x)$ satisfies the additional condition $\sup\limits_{x\in[0,1]}q(x)\leq \frac{π^{2}}{11}$, then $\frac{λ_{n}}{λ_{m}}\leq \frac{n^{2}% }{m^{2}}$ for $n>m\geq 1.$ For this result, we develop a new approach to study the monotonicity …
In this paper, we answer the various forms of nonnegative inverse eigenvalue problems with prescribed diagonal entries for order three: real or complex general matrices, symmetric stochastic matrices, and real or complex doubly stochastic matrices. We include the known cases, the symmetric matrices and real or complex stochastic matrices, to compare the other results and for completeness. In addition, for a given list of eigenvalues, we compute the exact range for the largest value of the diagonal entries of the various nonnegative matrices.
In this paper, we generalize some conclusions from the nonnegative irreducible tensor to the nonnegative weakly irreducible tensor and give more properties of eigenvalue problems.
Let $A$ be an irreducible (entrywise) nonnegative $n\times n$ matrix with eigenvalues $$ρ, b+ic,b-ic, λ_4,\cdots,λ_n,$$ where $ρ$ is the Perron eigenvalue. It is shown that for any $t \in [0, \infty)$ there is a nonnegative matrix with eigenvalues $$ρ+ \tilde t,λ_2+t,λ_3+t, λ_4 \cdots,λ_n,$$ whenever $\tilde t \ge γ_n t$ with $γ_3=1, γ_4 = 2, γ_5=\sqrt 5$ and $γ_n = 2.25$ for $n \ge 6$. The result improves that of Guo et al. Our proof depends on an auxiliary result in geometry asserting that the area of an $n$-sided convex polygon is bounded by $γ_n$ times the maximum area of the triangle lyin…
In this work, the real nonnegative inverse eigenvalue problem is solved for a particular class of permutative matrix. The necessary and sufficient condition there is also shown to be sufficient for the symmetric nonnegative inverse eigenvalue problem. A result due to Johnson and Paparella [MR3452738, Linear Algebra Appl. 493 (2016), 281--300] is extended to include normalized lists that satisfy the new sufficient condition.
In this paper we deal with circulant and partitioned into $n$-by-$n$ circulant blocks matrices and introduce spectral results concerning this class of matrices. The problem of finding lists of complex numbers corresponding to a set of eigenvalues of a nonnegative block matrix with circulant blocks is treated. Along the paper we call realizable list if its elements are the eigenvalues of a nonnegative matrix. The Guo's index $λ_0$ of a realizable list is the minimum spectral radius such that the list (up to the initial spectral radius) together with $λ_0$ is realizable. The Guo's index of block…
Let $A = [a_{i j}]_{i,j=1}^n$ be a nonnegative matrix with $a_{1 1} = 0$. We prove some lower bounds for the spread $s(A)$ of $A$ that is defined as the maximum distance between any two eigenvalues of $A$. If $A$ has only two distinct eigenvalues, then $s(A) \ge \frac{n}{2(n-1)} \, r(A)$, where $r(A)$ is the spectral radius of $A$. Moreover, this lower bound is the best possible.
The nonnegative inverse eigenvalue problem (NIEP) asks which lists of $n$ complex numbers (counting multiplicity) occur as the eigenvalues of some $n$-by-$n$ entry-wise nonnegative matrix. The NIEP has a long history and is a known hard (perhaps the hardest in matrix analysis?) and sought after problem. Thus, there are many subproblems and relevant results in a variety of directions. We survey most work on the problem and its several variants, with an emphasis on recent results, and include 130 references. The survey is divided into: a) the single eigenvalue problems; b) necessary conditions; …
We introduce the concept of shape partition of a tensor and formulate a general tensor eigenvalue problem that includes all previously studied eigenvalue problems as special cases. We formulate irreducibility and symmetry properties of a nonnegative tensor $T$ in terms of the associated shape partition. We recast the eigenvalue problem for $T$ as a fixed point problem on a suitable product of projective spaces. This allows us to use the theory of multi-homogeneous order-preserving maps to derive a new and unifying Perron-Frobenius theorem for nonnegative tensors which either implies earlier re…
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_…
On a closed weighted Riemannian manifold with nonnegative Bakry-Émery Ricci curvature, it is shown that the ratio of the $k$-th to first eigenvalues of the weighted Laplacian is dominated by $641k^2$, using an argument via the Cheeger constant. While improving the previous exponential upper bound, the order of $k$ here is optimal, and hence answers an open question of Funano. This approach works still on a compact finite-dimensional Alexandrov space of nonnegative curvature and proves affirmatively a conjecture of Funano and Shioya asserting a dimension free upper bound for eigenvalue ratios i…
It was conjectured by Escobar [J. Funct. Anal. 165 (1999), 101--116] that for an $n$-dimensional ($n\geq 3$) smooth compact Riemannian manifold with boundary, which has nonnegative Ricci curvature and boundary principal curvatures bounded below by $c>0$, the first nonzero Steklov eigenvalue is greater than or equal to $c$ with equality holding only on isometrically Euclidean balls with radius $1/c$. In this paper, we confirm this conjecture in the case of nonnegative sectional curvature. The proof is based on a combination of Qiu--Xia's weighted Reilly-type formula with a special choice of the…
Chao-Zhong Chen et al. $[{Proc}.$ ${Amer. Math. Soc},2013],$ proved the upper estimate $\frac{λ_{n}}{λ_{m}}\leq \frac{% n^{p}}{m^{p}}$ $ (n>m\geq 1) $ for Dirichlet Shrödinger operators with nonnegative and single-well potentials. In this paper we discuss the case of nonpositive potentials $q(x)$ continuous on the interval $[ 0,1] $. We prove that if $q(x)\leq 0$ and single-barrier then $\frac{λ_{n}}{λ_{m}}\geq \frac{n^{p}% }{m^{p}}$ for $λ_{n}>λ_{m}\geq -2q^{\ast },$ where $q^{\ast}=\inf\{q(0), q(1)\}$. Furthermore, we show that there exists $\ell_{0}\in ( 0,1] $ such that for all $\ell\in(0,…
For a nonnegative matrix A and real diagonal matrix D, two known inequalities on the spectral radius, r(A^2 D^2) >= r(AD)^2 and r(A) r(A D^2) >= r(AD)^2, leave open the question of what determines the order of r(A^2 D^2) with respect to r(A) r(A D^2). This is a special case of a broad class of problems that arise repeatedly in ecological and evolutionary dynamics. Here, sufficient conditions are found on A that determine orders in either direction. For a diagonally symmetrizable nonnegative matrix A with all positive eigenvalues and nonnegative D, r(A^2 D) <= r(A) r(AD). The reverse holds if a…
An invertible matrix is called a Perron similarity if one of its columns and the corresponding row of its inverse are both nonnegative or both nonpositive. Such matrices are of relevance and import in the study of the nonnegative inverse eigenvalue problem. In this work, Kronecker products of Perron similarities are examined and used to to construct ideal Perron similarities all of whose rows are extremal.
The symmetric nonnegative inverse eigenvalue problem (SNIEP) asks which sets of numbers (counting multiplicities) can be the eigenvalues of a symmetric matrix with nonnegative entries. While examples of such matrices are abundant in linear algebra and various applications, this question is still open for matrices of dimension $N\geq 5$. One of the approaches to solve the SNIEP was proposed by George W. Soules, relying on a specific type of eigenvectors (Soules vectors) to derive sufficient conditions for this problem. Elsner et al. later showed a canonical way to construct all Soules vectors, …
In this paper, the generalized eigenvalue complementarity problem for tensors (GEiCP-T) is addressed, which arises from the stability analysis of finite dimensional mechanical systems and find applications in differential dynamical systems. The general properties of the (GEiCP-T) have been studied. We establish its relationship with the generalized tensor eigenvalue problem. It follows that if exist, the number of $λ$-solutions can be bounded. We also give some sufficient conditions for the existence of the solution. In particular, there exists a unique solution of the (EiCP-T) (i.e., $J=[n]$)…
The paper is devoted to optimization of resonances associated with 1-D wave equations in inhomogeneous media. The medium's structure is represented by a nonnegative function B. The problem is to design for a given $α\in \R$ a medium that generates a resonance on the line $α+ ı\R$ with a minimal possible modulus of the imaginary part. We consider an admissible family of mediums that arises in a problem of optimal design for photonic crystals. This admissible family is defined by the constraints $0\leq b_1 \leq B (x) \leq b_2$ with certain constants $b_{1,2}$. The paper gives an accurate definit…
