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A list Λ = { λ 1 , ... , λ n } of complex numbers is said to be realizable if there exists an n × n nonnegative matrix A whose spectrum is Λ. In this case A is called a realizing matrix for Λ. The problem of characterizing all realizable lists Λ is known as the Nonnegative Inverse Eigenvalue Problem (NIEP). If A is required to be persymmetric , the Persymmetric Nonnegative Inverse Eigenvalue Problem (PNIEP) arises. In this paper, we show that the NIEP and the PNIEP are equivalent for any realizable list of three complex numbers. Furthermore, we give a new sufficient condition for a list of three complex numbers to be the spectrum of a persymmetric nonnegative matrix with prescribed diagonal entries. For some realizable lists of four complex numbers, we show that the NIEP and the PNIEP are equivalent. In general, we show that these problems are different. Finally, we found a sufficient condition for a trace zero list of five complex numbers to be the spectrum of a persymmetric nonnegative matrix. [ABSTRACT FROM AUTHOR]

An n-by-n real symmetric matrix $ H=[h_{i,j}] $ H = [ h i , j ] is said to be a Hankel matrix if $ h_{i,j}=h_{i-1,j+1} $ h i , j = h i − 1 , j + 1 , for each $ i=2,\ldots,n $ i = 2 , ... , n and $ j=1,\ldots,n-1 $ j = 1 , ... , n − 1. The symmetric nonnegative inverse eigenvalue problem (SNIEP) asks when a list $ \Lambda =\{\lambda _{1},\ldots,\lambda _{n}\} $ Λ = { λ 1 , ... , λ n } of real numbers is the spectrum of an n-by-n symmetric nonnegative matrix H. In this paper, we search for conditions on the list $ \Lambda =\{\lambda _{1},\ldots,\lambda _{n}\} $ Λ = { λ 1 , ... , λ n } for the matrix H to be Hankel. For n = 3, sufficient conditions are established. In particular, a necessary and sufficient condition is obtained if Λ is a list of three nonnegative numbers. Also, if $ \sum _{i=1}^{n}\lambda _{i}=0 $ ∑ i = 1 n λ i = 0 , we give conditions for realizability by a Hankel matrix. Finally, we present a special type of list that can serve as the spectrum of a Hankel nonnegative matrix with positive trace. Several of our results are constructive and provide a Hankel realizing matrix. [ABSTRACT FROM AUTHOR]

The notion of automatic selfadjointness of all ideals in a multiplicative semigroup of the bounded linear operators on a separable Hilbert space B (H) arose in a 2015 discussion with Heydar Radjavi who pointed out that B (H) and the finite rank operators F (H) possessed this unitary invariant property which category we named SI semigroups (for automatic selfadjoint ideal semigroups). Equivalent to the SI property is the solvability, for each A in the semigroup, of the bilinear operator equation A ⁎ = X A Y which we believe is a new connection relating the semigroup theory with the theory of operator equations. We found in our earlier works in the subject that even at the basic level of singly generated semigroups, the investigation of SI semigroups led to interesting algebraic and analytic phenomena when generated by rank one operators, normal operators, partial and power partial isometries, subnormal-hyponormal-essentially normal operators, and weighted shift operators; and generated by commuting families of normal operators. In this paper, we focus on a separate M n (C) treatment for singly generated SI semigroups that requires studying the solvability of the bilinear matrix equation A ⁎ = X A Y in a multiplicative semigroup of finite matrices. This separate focus is needed because the techniques employed in our earlier works we could not adapt to finite matrices. In this paper we find that for certain classes of generators, being a partial isometry is equivalent to generating an SI semigroup. Such classes are: degree 2 nilpotent matrices, weighted shifts, and non-normal Jordan matrices. For the key tools used to establish these equivalences, we developed a number of necessary conditions for singly generated semigroups to be SI for the very general classes: nonselfadjoint matrices, nonzero nilpotent matrices, nonselfadjoint invertible matrices, and Jordan blocks. We also show, for a nonselfadjoint matrix generator in an SI semigroup, the matrix being a partial isometry is equivalent to having norm one. And as an aside, we also prove necessary generator conditions for the SI property when generated by matrices with nonnegative entries. [ABSTRACT FROM AUTHOR]

A list Λ = {λ1, λ2, ..., λn} of complex numbers is said to be realizable if it is the spectrum of a nonnegative matrix. Λ is said to be universally realizable (UR) if it is realizable for each possible Jordan canonical form allowed by Λ. In this paper, using companion matrices and applying a procedure by Šmigoc, we provide sufficient conditions for the universal realizability of left half-plane spectra, that is, spectra Λ = {λ1, ..., λn} with λ1 > 0, Re λi ≤ 0, i = 2, ..., n. It is also shown how the effect of adding a negative real number to a not UR left half-plane list of complex numbers, makes the new list UR, and a family of left half-plane lists that are UR is characterized. [ABSTRACT FROM AUTHOR]

In this work, doubly stochastic combined matrices are studied. The combined matrix is known as Relative Gain Array in control theory. In particular, the characterization of all matrices such that their combined matrix is doubly stochastic is studied for real matrices of order 3. In addition, we give necessary and sufficient conditions so that three positive real numbers are the diagonal entries of a doubly stochastic matrix. [ABSTRACT FROM AUTHOR]

The independence number of a square matrix A , denoted by α (A) , is the maximum order of its principal zero submatrices. Given any integer n , let S n + be the set of n × n nonnegative symmetric matrices with zero diagonal. We characterize the linear maps on S n + preserving the independence number of all matrices in S n +. [ABSTRACT FROM AUTHOR]

New bounds for the Perron root ρ (A) of a positive matrix A are proposed. Let R 1 ≥ R 2 ≥ ⋯ ≥ R n be the row sums of matrix A listed in descending order. We use a pairwise combination method to prove that min 1 ≤ k ≤ j { (1 − t) R k + t R n − k + 1 } ≤ ρ (A) ≤ max 1 ≤ k ≤ j { t R k + (1 − t) R n − k + 1 } , with j = ⌈ n / 2 ⌉ , t = M / (M + m) , and M (respectively m) is the maximum (resp. minimum) entry of A. This result improves the well-known bounds R n ≤ ρ (A) ≤ R 1 , and the same method can also be used to improve several other known bounds. [ABSTRACT FROM AUTHOR]

Methods for the algorithmic construction of stability indicators of nonnegative matrices is described, and the application of these indicators to problems of modern mathematical biology and epidemiology is discussed. Specific features of such indicators when they are applied to problems about the parametric loss of stability of trivial equilibrium states of discrete dynamical systems are pointed out. Estimates of the efficiency of algorithms based on the proposed methods for the case of systems determined by sparse matrices are given. Examples of using the constructed algorithms for such systems are discussed. [ABSTRACT FROM AUTHOR]

In this article, we expand a classical Frobenius' result upon consecutive k -th powers of nonnegative matrices to establish sequences of new lower and upper bounds for the spectral radius with respect to the positive integer k , each term of which is formulated by the average (k + 1) -row sums of the nonnegative matrix. With the aid of the average (k + 1) -row sums and taking the extreme entries of the matrix, we study new bounds generalizing existing formulae and we produce sequences of new tighter approximations for the spectral radius. The monotonicity and convergence properties of the constructed sequences are explored and certain conditions are stated under which the new bounds are sharper than Frobenius' bounds and other existing formulae. We further characterize the cases of equality in the aforesaid bounds, when the matrix is irreducible. Throughout, we perform illustrative numerical examples to showcase the efficiency of our proposed bounds and make comparisons among them. [ABSTRACT FROM AUTHOR]

An antinorm is a concave nonnegative homogeneous functional on a convex cone. It is shown that if the cone is polyhedral, then every antinorm has a unique continuous extension from the interior of the cone. The main facts of the duality theory in convex analysis, in particular, the Fenchel-Moreau theorem, are generalized to antinorms. However, it is shown that the duality relation for antinorms is discontinuous. In every dimension, there are infinitely many self-dual antinorms on the positive orthant and, in particular, infinitely many autopolar polyhedra. For the two-dimensional case, we characterize them all. The classification in higher dimensions is left as an open problem. Applications to linear dynamical systems, to the Lyapunov exponent of random matrix products, to the lower spectral radius of nonnegative matrices, and to convex trigonometry are considered. [ABSTRACT FROM AUTHOR]

An n-by-n real symmetric matrix H = [ h i , j ] is said to be a Hankel matrix if h i , j = h i − 1 , j + 1 , for each i = 2 , … , n and j = 1 , … , n − 1 . The symmetric nonnegative inverse eigenvalue problem (SNIEP) asks when a list Λ = { λ 1 , … , λ n } of real numbers is the spectrum of an n-by-n symmetric nonnegative matrix H. In this paper, we search for conditions on the list Λ = { λ 1 , … , λ n } for the matrix H to be Hankel. For n = 3, sufficient conditions are established. In particular, a necessary and sufficient condition is obtained if Λ is a list of three nonnegative numbers. Also, if ∑ i = 1 n λ i = 0 , we give conditions for realizability by a Hankel matrix. Finally, we present a special type of list that can serve as the spectrum of a Hankel nonnegative matrix with positive trace. Several of our results are  constructive and provide a Hankel realizing matrix. [ABSTRACT FROM AUTHOR]

A fractional matching of a graph G is a function f:E(G) → [0, 1] such that for any v ∈ V(G), ∑ e ∈ E G (v) f (e) ≤ 1 where EG(v) = {e ∈ E(G): e is incident with v in G}. The fractional matching number of G is μ f (G) = max { ∑ e ∈ E (G) f (e) : f is a fractional matching of G}. For any real numbers a ≥ 0 and k ∈ (0, n), it is observed that if n = |V(G)| and δ (G) > n − k 2 , then μ f (G) > n − k 2 . We determine a function φ(a, n, δ, k) and show that for a connected graph G with n = |V(G)|, δ (G) ≤ n − k 2 , spectral radius λ1(G) and complement G ¯ , each of the following holds. If λ1 (aD(G) + A(G)) < φ(a, n, δ, k), then μ f (G) > n − k 2. If λ 1 (a D (G ¯) + A (G ¯)) < (a + 1) (δ + k − 1) , then μ f (G) > n − k 2. As applications, we prove a relationship between μf(G) and λ1(aD(G) + A(G)) for a graph G. Furthermore, sufficient spectral conditions for a graph to have a fractional perfect matching are also obtained. [ABSTRACT FROM AUTHOR]

In this survey paper, we consider the problem of which nonempty bounded convex subset of the complex plane is the numerical range of some bounded linear operator on a complex separable Hilbert space. We start in Section 1 with general operators and move subsequently to operators in certain special classes such as normal and hyponormal operators in Section 2, Toeplitz operators in Section 3, Hankel operators in Section 4, compact operators in Section 5, nilpotent operators and roots of identity in Section 6, Sn-matrices and companion matrices in Section 7, and, finally, nonnegative matrices in Section 8. The known main results will be briefly sketched, which are interspersed with relevant unsolved problems. [ABSTRACT FROM AUTHOR]

This article considers two inverse eigenvalue problems for Lefkovitch and doubly Lefkovitch matrices. The problems consist of constructing these matrices from the maximal eigenvalues of all its leading principal submatrices, eigenvectors associated with some of these eigenvalues, and some diagonal entries. We give sufficient conditions for the existence of such matrices. In particular, when the matrices have constant diagonal entries. In addition, we provide numerical examples to show the results obtained. [ABSTRACT FROM AUTHOR]

We give a family of counterexamples of a theorem on a new upper bound for the α-indices of graphs in the paper that is mentioned in the title. We also provide a new upper bound for corrigendum. [ABSTRACT FROM AUTHOR]

We say that a list Λ = { λ 1 , ... , λ n } of complex numbers is realizable, if it is the spectrum of a nonnegative matrix A (a realizing matrix). We say that Λ is universally realizable if it is realizable for each possible Jordan canonical form allowed by Λ. This work studies the universal realizability of spectra in low dimension, that is, realizable spectra of size n ≤ 5. It is clear that for n ≤ 3 the concepts of universally realizable and realizable are equivalent. The case n = 4 is easily deduced from previous results in [7]. We characterize the universal realizability of real spectra of size 5 and trace zero, and we describe a region for the universal realizability of nonreal 5-spectra with trace zero. As an important by-product of our study, we also show that realizable lists on the left half-plane, that is, lists Λ = { λ 1 , ... , λ n } , where λ 1 is the Perron eigenvalue and Re λ i ≤ 0 , for i = 2 , ... , n , are not necessarily universally realizable. [ABSTRACT FROM AUTHOR]

Given a square nonnegative matrix , a simple algorithm is suggested for constructing a stability indicator characterizing the localization of its spectrum in the unit disk. Theorems are proved which establish the possibility of using the maximum of over all possible principal submatrices of as a suitable indicator and give conditions under which such a maximum can be calculated only over a certain chain of leading principal submatrices. Applied problems that need such constructions and a relationship between the obtained results and similar results established for a number of matrices of special form are considered. [ABSTRACT FROM AUTHOR]

The list Λ = { λ 1 , λ 2 , ... , λ n } , of complex numbers, is said to be realizable if it is the spectrum of an entrywise nonnegative matrix A. If A is diagonalizable, Λ is said to be diagonalizably realizable. Λ is universally realizable if it is realizable for each possible Jordan canonical form allowed by Λ. Here, new realizability criteria are introduced. A diagonalizable version of a perturbation result by Rado is also proved. It allows to construct diagonalizable nonnegative matrices with prescribed spectrum. Criteria to decide the universal realizability of spectra are also established. [ABSTRACT FROM AUTHOR]

In this paper, we propose some sharper upper bounds for the spectral radius of the Hadamard product of two nonnegative matrices. These bounds improve the existing results. In addition, examples are also given to show that the new upper bounds can be achieved. [ABSTRACT FROM AUTHOR]

Let k ≥ 2 be an integer. If a square 0-1 matrix A satisfies A k = A , then A is said to be k -idempotent. In this paper, we give a characterization of k -idempotent 0-1 matrices. We also determine the maximum number of nonzero entries in k -idempotent 0-1 matrices of a given order as well as the k -idempotent 0-1 matrices attaining this maximum number. [ABSTRACT FROM AUTHOR]

We say that a list Λ = { λ 1 , λ 2 , ... , λ n } of complex numbers is realizable if it is the spectrum of an entrywise nonnegative matrix A. We say that Λ is universally realizable (UR) if Λ is realizable for each possible Jordan canonical form allowed by Λ. The universal realizability problem of spectra (URP) has been solved for certain kind of real and complex spectra, mainly spectra in the left half-plane. In this paper we look for extending the kind of spectra that are UR by given new sufficient conditions for the URP to have a solution. In particular, we show that some realizability criteria for the real and symmetric nonnegative inverse eigenvalue problem are realizability criteria for the URP as well. [ABSTRACT FROM AUTHOR]

We develop a matrix approach to the maximal acyclic subgraph (MAS) problem by reducing it to find the closest nilpotent matrix to the matrix of the graph. Using recent results on the closest Schur stable systems and on minimizing the spectral radius over special sets of nonnegative matrices, we obtain an algorithm for finding an approximate solution of the MAS problem. Numerical results for graphs from 50 to 1500 vertices demonstrate its fast convergence and give a rate of approximation larger than 0.6 in most cases. This method also gives the precise solution for the following weakened version of MAS: find the minimal r such that the graph can be made acyclic by cutting at most r incoming edges from each vertex. We also consider several modifications in the case when each vertex is assigned its own maximal number ri of cut edges, and some of the edges are untouchable."" Some applications are discussed. [ABSTRACT FROM AUTHOR]

In this paper, we deal with Leslie and doubly Leslie matrices of order n. In particular, with the companion and doubly companion matrices. We study three inverse eigenvalues problems which consist of constructing these matrices from the maximal eigenvalues of its all leading principal submatrices. For Leslie and doubly companion matrices, an eigenvector associated with the maximal eigenvalue of the matrix is additionally considered, and for the doubly Leslie matrix also an eigenvector associated with the maximal eigenvalue of leading principal submatrix of order n − 1 is required. We give necessary and sufficient conditions for the existence of a Leslie matrix and a companion matrix, and sufficient conditions for the existence of a doubly Leslie matrix and a doubly companion matrix. Our results are constructive and generate an algorithmic procedure to construct these special kinds of matrices. [ABSTRACT FROM AUTHOR]

A g -circulant matrix A , is defined as a matrix of order n where the elements of each row of A are identical to those of the previous row, but are moved g positions to the right and wrapped around. Using number theory, certain spectra of g -circulant real matrices are given explicitly. The obtained results are applied to Nonnegative Inverse Eigenvalue Problem to construct nonnegative, g -circulant matrices with given appropriated spectrum. Additionally, some g -circulant matrices are reconstructed from its main diagonal entries. [ABSTRACT FROM AUTHOR]

A general method is given for merging blocks in the Jordan canonical form of a nonnegative matrix. As a consequence, results, more general than any prior ones, are given for the universal realizability of spectra, that is, spectra which are realizable by a nonnegative matrix for each possible Jordan canonical form allowed by the spectrum. In particular, we generalize a classical result due to Minc, regarding positive diagonalizable matrices. For example, any spectrum that is diagonalizably realizable by a nonnegative matrix with mostly positive off-diagonal entries is universally realizable. [ABSTRACT FROM AUTHOR]

A list Λ = { λ 1 , λ 2 , ... , λ n } of complex numbers is said to be realizable if it is the spectrum of an entrywise nonnegative matrix. In this paper we intent to characterize those lists of complex numbers, which are realizable by a centrosymmetric nonnegative matrix. In particular, we show that lists of nonnegative real numbers, and lists of complex numbers of Suleimanova type (except in one particular case), are always the spectrum of some centrosymmetric nonnegative matrix. For the general lists we give sufficient conditions via a perturbation result. We also show that for n = 4 , every realizable list of real numbers is also realizable by a nonnegative centrosymmetric matrix. [ABSTRACT FROM AUTHOR]

In this paper we present spectral results for matrices partitioned into higher order blocks, where not all blocks are necessarily square. Using these results sufficient conditions on a given list to be the list of eigenvalues of a nonnegative matrix are obtained and the corresponding matrix is constructed. In particular, spectral results for symmetric matrices are derived. A result on Guo perturbations for permutative matrices on given lists is generalized. [ABSTRACT FROM AUTHOR]

The method of constructing a stability indicatrix of a nonnegative matrix having the form of a polynomial of its coefficients is presented. The algorithm of construction and conditions of its applicability are specified. The applicability of the algorithm is illustrated on examples of constructing the stability indicatrix for a series of functions widely used in simulation of the dynamics of discrete biological communities, for solving evolutionary optimality problems arising in biological problems of evolutionary selection, for identification of the conditions of the pandemic in a distributed host population. [ABSTRACT FROM AUTHOR]

Abstract A list Λ = { λ 1 , λ 2 , ... , λ n } of complex numbers is said to be realizable if it is the spectrum of an entrywise nonnegative matrix. The list Λ is said to be universally realizable (UR) if it is the spectrum of a nonnegative matrix for each possible Jordan canonical form allowed by Λ. It is well known that an n × n nonnegative matrix A is co-spectral to a nonnegative matrix B with constant row sums. In this paper, we extend the co-spectrality between A and B to a similarity between A and B , when the Perron eigenvalue is simple. We also show that if ϵ ≥ 0 and Λ = { λ 1 , λ 2 , ... , λ n } is UR , then { λ 1 + ϵ , λ 2 , ... , λ n } is also UR. We give counter-examples for the cases: Λ = { λ 1 , λ 2 , ... , λ n } is UR implies { λ 1 + ϵ , λ 2 − ϵ , λ 3 , ... , λ n } is UR , and Λ 1 , Λ 2 are UR implies Λ 1 ∪ Λ 2 is UR. [ABSTRACT FROM AUTHOR]

Abstract The restricted version of the overlapping Schur complement (SchurRAS) preconditioner was introduced by Li and Saad (2006) for the solution of linear system A x = b , and numerical results have shown that the SchurRAS method outperforms the restricted additive Schwarz (RAS) method both in terms of iteration count and CPU time. In this paper, based on meticulous derivation, we give an algebraic representation of the SchurRAS preconditioner, and prove that the SchurRAS method is convergent under the condition that A is an M -matrix and it converges faster than the RAS method. [ABSTRACT FROM AUTHOR]

We consider the problem of computing the closest stable/unstable nonnegative matrix to a given real matrix. The distance between matrices is measured in the Frobenius norm. The problem is addressed for two types of stability: the Schur stability (the matrix is stable if its spectral radius is smaller than one) and the Hurwitz stability (the matrix is stable if its spectral abscissa is negative). We show that the closest unstable matrix can always be explicitly found. The problem of computing the closest stable matrix to a nonnegative matrix is a hard problem even if the stable matrix is not constrained to be nonnegative. Adding the nonnegativity constraint makes the problem even more difficult. For the closest stable matrix, we present an iterative algorithm which converges to a local minimum with a linear rate. It is shown that the total number of local minima can be exponential in the dimension. Numerical results and the complexity estimates are presented. [ABSTRACT FROM AUTHOR]

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 circulant matrices with circulant blocks is obtained, and in consequence, necessary and sufficient conditions concerning the NIEP, Nonnegative Inverse Eigenvalue Problem, for the realizability of some spectra are given. [ABSTRACT FROM AUTHOR]

A square matrix of order n with n ≥ 2 is called permutative matrix when all its rows are permutations of the first row. In this paper recalling spectral results for partitioned into 2-by-2 symmetric blocks matrices sufficient conditions on a given complex list to be the list of the eigenvalues of a nonnegative permutative matrix are given. In particular, we study NIEP and PNIEP when some complex elements in the lists under consideration have non-zero imaginary part. Realizability regions for nonnegative permutative matrices are obtained. A Guo's realizability-preserving perturbations result is obtained. [ABSTRACT FROM AUTHOR]

For a nonnegative square matrix, a method of constructing a stability indicator in the form of a polynomial in matrix coefficients is presented. Additionally, an algorithm for indicator calculation is described and the relationship of the resulting indicator with determinants of some matrices built on the basis of the original one is demonstrated. [ABSTRACT FROM AUTHOR]

A list of eigenvalues is said to be realizable if it is the spectrum of a nonnegative matrix, diagonalizably realizable (DR) if it is the spectrum of a diagonalizable nonnegative matrix, and universally realizable (UR) if there is a nonnegative matrix with this spectrum and any possible Jordan canonical form allowed by the spectrum. The nonnegative inverse eigenvalue problem (NIEP) asks which lists are realizable. It is known that there exist spectra that are realizable, but not DR. We raise the question of whether DR implies UR. This is known in a few cases, including n ≤ 4 , nonnegative spectra, and Suleimanova spectra. We add some new classes of spectra that are UR. [ABSTRACT FROM AUTHOR]

This paper explicitly details the relation between M -matrices, nonnegative roots of nonnegative matrices, and the embedding problem for finite-state stationary Markov chains. The set of nonsingular nonnegative matrices with arbitrary nonnegative roots is shown to be the closure of the set of matrices with matrix roots in IM . The methods presented here employ nothing beyond basic matrix analysis, however it answers a question regarding M -matrices posed over 30 years ago and as an application, a new characterization of the set of all embeddable stochastic matrices is obtained as a corollary. [ABSTRACT FROM AUTHOR]

Let $A_{1}, A_{2},\ldots, A_{k}$ be nonnegative matrices. In this paper, some upper bounds for the spectral radius $\rho(A_{1}\circ A_{2}\circ\cdots\circ A_{k})$ are proposed. These bounds generalize some existing results, and comparisons between these bounds are also considered. [ABSTRACT FROM AUTHOR]

A square matrix of order n with n ≥ 2 is called a permutative matrix or permutative when all its rows (up to the first one) are permutations of precisely its first row. In this paper, the spectra of a class of permutative matrices are studied. In particular, spectral results for matrices partitioned into 2-by-2 symmetric blocks are presented and, using these results sufficient conditions on a given list to be the list of eigenvalues of a nonnegative permutative matrix are obtained and the corresponding permutative matrices are constructed. Guo perturbations on given lists are exhibited. [ABSTRACT FROM AUTHOR]

In this paper, we give some new sharp upper and lower bounds for the spectral radius of a nonnegative irreducible matrix. Using these bounds, we obtain some new and improved bounds for the signless Laplacian spectral radius of a graph or a digraph. [ABSTRACT FROM AUTHOR]

New bounds for the Perron root ρ ( A ) of a nonnegative matrix A are proposed. We prove that min 1 ≤ i ≤ n ⁡ r i ( A B ) r i ( B ) ≤ ρ ( A ) ≤ max 1 ≤ i ≤ n ⁡ r i ( A B ) r i ( B ) where B is an arbitrary matrix with row sums r i ( B ) = ∑ k b i k > 0 , i = 1 , . . . , n . The bounds of H. Minc [1] and Shulin Liu [6] are both special cases of this result. And based on this result, we also get some bounds for the spectral radius of iteration matrices. [ABSTRACT FROM AUTHOR]

We give sharp upper and lower bounds for the spectral radius of a nonnegative matrix with positive row sums using average 3-row sums, compare these bounds with the existing bounds using the average 2-row sums by examples, and apply them to the adjacency matrix and the signless Laplacian matrix of a digraph or a graph. [ABSTRACT FROM AUTHOR]

A list of complex numbers is realizable if it is the spectrum of a nonnegative matrix. In 1949 Suleı̌manova posed the nonnegative inverse eigenvalue problem (NIEP): the problem of determining which lists of complex numbers are realizable. The version for reals of the NIEP (RNIEP) asks for realizable lists of real numbers. In the literature there are many sufficient conditions or criteria for lists of real numbers to be realizable. We will present an unified account of these criteria. Then we will see that the decision problem associated to the RNIEP is NP-hard and we will study the complexity for the decision problems associated to known criteria. [ABSTRACT FROM AUTHOR]

It is known that the critical exponent (CE) for conventional, continuous powers ofn-by-ndoubly nonnegative (DN) matrices is. Here, we consider the larger class of diagonalizable, entrywise nonnegativen-by-nmatrices with nonnegative eigenvalues (generalized doubly nonnegative (GDN)). We show that, again, a CE exists and is able to bind with a low-coefficient quadratic. However, the CE is larger than in the DN case; in particular, 2 for. There seems to be a connection with the index of primitivity, and a number of other observations are made and questions raised. It is shown that there is no CE for continuous Hadamard powers of GDN matrices, despite it also beingfor DN matrices. [ABSTRACT FROM AUTHOR]

In this paper, we obtain the sharp upper and lower bounds for the spectral radius of a nonnegative irreducible matrix. We also apply these bounds to various matrices associated with a graph or a digraph, obtain some new results or known results about various spectral radii, including the adjacency spectral radius, the signless Laplacian spectral radius, the distance spectral radius, the distance signless Laplacian spectral radius of a graph or a digraph. [ABSTRACT FROM AUTHOR]

We propose an inverse iterative method for computing the Perron pair of an irreducible nonnegative third order tensor. The method involves the selection of a parameter θk in the kth iteration. For every positive starting vector, the method converges quadratically and is positivity preserving in the sense that the vectors approximating the Perron vector are strictly positive in each iteration. It is also shown that θk = 1 near convergence. The computational work for each iteration of the proposed method is less than four times (three times if the tensor is symmetric in modes two and three, and twice if we also take the parameter to be 1 directly) that for each iteration of the Ng-Qi-Zhou algorithm, which is linearly convergent for essentially positive tensors. [ABSTRACT FROM AUTHOR]

The combined matrix of a nonsingular matrix A is the Hadamard (entry wise) product C ( A ) = A ∘ ( A − 1 ) T . Since each row and column sum of C ( A ) is equal to one, the combined matrix is doubly stochastic when it is nonnegative. In this work, we study the nonnegativity of the combined matrix of sign regular matrices, based upon their signature. In particular, a few coordinates of the signature ε of A play a crucial role in determining whether or not C ( A ) is nonnegative. [ABSTRACT FROM AUTHOR]

By the square root of a (square) matrix A we mean a matrix B that satisfies B 2 = A . In this paper, we begin a study of the (entrywise) nonnegative square roots of nonnegative matrices, adopting mainly a graph-theoretic approach. To start with, we settle completely the question of existence and uniqueness of nonnegative square roots for 2-by-2 nonnegative matrices. By the square of a digraph H , denoted by H 2 , we mean the digraph with the same vertex set as H such that ( i , j ) is an arc if there is a vertex k such that ( i , k ) and ( k , j ) are both arcs in H . We call a digraph H a square root of a digraph G if H 2 = G . It is observed that a necessary condition for a nonnegative matrix to have a nonnegative square root is that its digraph has a square root, and also that a digraph G has a square root if and only if there exists a nonnegative matrix A with digraph G such that A has a nonnegative square root. We consider when or whether certain kinds of digraphs (including digraphs that are disjoint union of directed paths and circuits, permutation digraphs or a special kind of bigraphs) have square roots. We also consider when certain kinds of nonnegative matrices (including monomial matrices, rank-one matrices and nilpotent matrices with index two) have nonnegative square roots. A known characterization of loopless digraphs to have square roots, due to F. Escalantge, L. Montejano, and T. Rojano, is extended (and amended) to digraphs possibly with loops. Some natural open questions are also posed. [ABSTRACT FROM AUTHOR]