Nonnegative Inverse Eigenvalue Problem Research Articles

The nonnegative inverse eigenvalue problem with prescribed zero patterns in dimension three

The nonnegative inverse eigenvalue problem is considered in this paper with the additional restriction of fixed zero patterns in the matrix. A full analysis of the $3\times 3$ case is given. Some remarks on the four-dimensional case are made.

About the persymmetric nonnegative inverse eigenvalue problem in low dimension

A list Λ={λ1,…,λn} 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.

A characterization of sets realizable by compensation in the SNIEP

The symmetric nonnegative inverse eigenvalue problem (SNIEP) is the problem of characterizing all possible spectra of entry-wise nonnegative symmetric matrices of given dimension. A list of real numbers is said to be symmetrically realizable if it is the spectrum of some nonnegative symmetric matrix. One of the most general sufficient conditions for realizability is the so-called C-realizability, which amounts to some kind of compensation between the positive and negative entries of the list of real numbers whose realizability one is trying to decide. A combinatorial characterization of C-realizable lists with zero trace was given in [11]. In this paper we make use of a recursive method for constructing simmetrically realizable lists due to Ellard and Šmigoc [3] to extend this combinatorial characterization of C-realizability to general lists with nonnegative trace. One consequence of this characterization is that the set of nonnegative C-realizable lists is a union of polyhedral cones whose faces are described by equations involving only linear combinations with coefficients 1 and −1 of the entries in the list. Another remarkable consequence is the monotonicity of C-realizability, i.e., the operation of increasing any positive entry of a C-realizable list preserves C-realizability.

Proof of a conjecture on polynomials preserving nonnegative matrices

We consider polynomials in R[x] which map the set of nonnegative matrices of a given order into itself. Let n be a positive integer and define Pn:={p∈R[x]:p(A)≥0,forallA≥0,A∈Rn,n}. This set plays a role in the Nonnegative Inverse Eigenvalue Problem. Clark and Paparella conjectured that Pn+1 is strictly contained in Pn. We prove this conjecture. The structure of Pn is also of interest, and for this purpose define, for any positive integer m, Pn,m={p∈R[x]:degree(p)≤mandp(A)≥0,forallA≥0,A∈Rn,n}. This is a convex cone. For 0<m<2n it is known to be polyhedral, and the question is whether for m≥2n it is not polyhedral. We answer it in the affirmative when n=2.

Realizable regions for the symmetric nonnegative inverse eigenvalue problem for 6 × 6 matrices

The symmetric nonnegative inverse eigenvalue problem is to determine when a multiset of n real numbers is the spectrum of a symmetric nonnegative n × n matrix. The problem is solved only for n ≤ 4 . In this article, the authors examine the set of possible spectra for nonnegative symmetric 6 × 6 matrices, dividing the set into points that are realizable, points that are unrealizable by known necessary conditions, and points that satisfy all known necessary conditions, but their realizability remains unknown.

A note for the SNIEP in size 5

The purpose of this note is to establish the current state of the knowledge about the SNIEP (symmetric nonnegative inverse eigenvalue problem) in size 5 with just one repeated eigenvalue.

Riemannian linearized proximal algorithms for nonnegative inverse eigenvalue problem

Riemannian linearized proximal algorithms for nonnegative inverse eigenvalue problem

On Hankel matrices and the symmetric nonnegative inverse eigenvalue problem

An n-by-n real symmetric matrix is said to be a Hankel matrix if , for each and . The symmetric nonnegative inverse eigenvalue problem (SNIEP) asks when a list 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 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 , 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.

Kronecker products of Perron similarities

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 construct ideal Perron similarities all of whose rows are extremal.

Polynomials that preserve nonnegative matrices

In further pursuit of a solution to the celebrated nonnegative inverse eigenvalue problem, Loewy and London ((1978/1979) [8]) posed the problem of characterizing all polynomials that preserve all nonnegative matrices of a fixed order. If Pn 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 Pn. However, it is known that Pn contains polynomials with negative entries. In this work, novel results for Pn with respect to the coefficients of the polynomials belonging to Pn. 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.

The Non-negative Inverse Eigenvalue Problem for Tridiagonal Matrix, Circulant Matrix and Symmetric Matrix

The Non-negative Inverse Eigenvalue Problem (NIEP) is a sub-problem extracted from inverse eigenvalue problem with a long history from 1930s determining the sufficient and necessary conditions in order that, σ={ λ 1,…, λ n }to be the spectrum of an entry wise non-negative n x n matrix. There had been many excellent researchers and scholars who contributed to discover many practical theories. The following of this paper would be then separated into two parts to further analyze the NIEP. In the first section of the paper, some important preexisting conclusions would be demonstrated and the groups’ understanding of these indispensable theories would be expressed. In the second section, three special and wildly used matrix, including tridiagongal matrix, circulant matrix, and symmetric matrix would be considered. The solution of the NIEP of these matrices done by the group would be expressed.

Some additional notes on the spectra of non-negative symmetric 5 x 5 matrices

The Symmetric Non-negative Inverse Eigenvalue Problem (SNIEP) asks when is a list $ \sigma = \left( \lambda_{1}, \lambda_{2}, \dots, \lambda_{n} \right) $ of real, monotonically decreasing numbers, the spectrum of an $n \times n$, symmetric, non-negative matrix $A$. In that case, we say $\sigma$ is realizable and $A$ is a realizing matrix. Here, we consider the case $n=5$, the lowest value of $n$ for which the problem is unsolved. Let $ s_{1}(\sigma) = \sum_{i=1}^5 \lambda_{i} $ and $ s_{3}(\sigma) = \sum_{i=1}^5 {\lambda_{i}}^3 $. It is known that to complete the solution for $n=5$, it remains to consider the case $\lambda_{3} &gt; s_{1}(\sigma)$, so let $y=\lambda_{3}- s_{1}(\sigma)$ and assume $y \geq 0$. We prove that if $\sigma$ is realizable, then $s_{3}(\sigma) \geq s_{1}(\sigma)^3+6s_{1}(\sigma)y(s_{1}(\sigma)+y)$. This strengthens the inequality $s_{3}(\sigma) \geq s_{1}(\sigma)^3$ obtained by Loewy and Spector, which in turn strengthens the inequality $ 25s_{3}(\sigma) \geq s_{1}(\sigma)^3 $, one of the Johnson--Loewy--London inequalities. As an application of the new inequality, we show that certain lists previously unknown as far as their realizability is concerned are not realizable.

A characterization of trace-zero sets realizable by compensation in the SNIEP

The symmetric nonnegative inverse eigenvalue problem (SNIEP) is the problem of characterizing all possible spectra of entry-wise nonnegative symmetric matrices of given dimension. A list of real numbers is said to be symmetrically realizable if it is the spectrum of some nonnegative symmetric matrix. One of the most general sufficient conditions for realizability is so-called C-realizability, which amounts to some kind of compensation between positive and negative entries of the list. In this paper we present a combinatorial characterization of C-realizable lists with zero sum, together with explicit formulas for C-realizable lists having at most four positive entries. One of the consequences of this characterization is that the set of zero-sum C-realizable lists is shown to be a union of polyhedral cones whose faces are described by equations involving only linear combinations with coefficients 1 and −1 of the entries in the list.

The NIEP and the positive realization problem

The nonnegative inverse eigenvalue problem is the problem of determining necessary and sufficient conditions for a multiset of complex numbers to be the spectrum of a nonnegative real matrix of size equal to the cardinality of the multiset itself. The problem is longstanding and proved to be very difficult so that several variations have been defined by considering particular classes of multisets and nonnegative real matrices. In this paper, a novel variation of the problem is proposed. This variation is motivated by a practical application in the positive realization problem, that is the problem of characterizing existence and minimality of a positive state--space representation of a given transfer function.

A Note on NIEP for Leslie and Doubly Leslie Matrices

The nonnegative inverse eigenvalue problem (NIEP) consists of finding necessary and sufficient conditions for the existence of a nonnegative matrix with a given list of complex numbers as its spectrum. If the matrix is required to be Leslie (doubly Leslie), the problem is called the Leslie (doubly Leslie) nonnegative eigenvalue inverse problem. In this paper, necessary and/or sufficient conditions for the existence and construction of Leslie and doubly Leslie matrices with a given spectrum are considered.

On the universal realizability problem

We say that a list Λ={λ1,λ2,…,λn} 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.

On the spectra of some g-circulant matrices and applications to nonnegative inverse eigenvalue problem

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.

The inverse eigenvalue problem for Leslie matrices

The Nonnegative Inverse Eigenvalue Problem (NIEP) is the problem of determining necessary and sufficient conditions for a list of $n$ complex numbers to be the spectrum of an entry--wise nonnegative matrix of dimension $n$. This is a very difficult and long standing problem and has been solved only for $n\leq 4$. In this paper, the NIEP for a particular class of nonnegative matrices, namely Leslie matrices, is considered. Leslie matrices are nonnegative matrices, with a special zero--pattern, arising in the Leslie model, one of the best known and widely used models to describe the growth of populations. The lists of nonzero complex numbers that are subsets of the spectra of Leslie matrices are fully characterized. Moreover, the minimal dimension of a Leslie matrix having a given list of three numbers among its spectrum is provided. This result is partially extended to the case of lists of $n &gt; 2$ real numbers.

Brauer's theorem and nonnegative matrices with prescribed diagonal entries

The problem of the existence and construction of nonnegative matrices with prescribed eigenvalues and diagonal entries is an important inverse problem, interesting by itself, but also necessary to apply a perturbation result, which has played an important role in the study of certain nonnegative inverse spectral problems. A number of partial results about the problem have been published by several authors, mainly by H. \v{S}migoc. In this paper, the relevance of a Brauer's result, and its implication for the nonnegative inverse eigenvalue problem with prescribed diagonal entries is emphasized. As a consequence, given a list of complex numbers of \v{S}migoc type, or a list $\Lambda = \left\{\lambda _{1},\ldots ,\lambda _{n} \right \}$ with $\operatorname{Re}\lambda _{i}\leq 0,$ $\lambda _{1}\geq -\sum\limits_{i=2}^{n}\lambda _{i}$, and $\left\{-\sum\limits_{i=2}^{n}\lambda _{i},\lambda _{2},\ldots ,\lambda _{n} \right\}$ being realizable; and given a list of nonnegative real numbers $% \Gamma = \left\{\gamma _{1},\ldots ,\gamma _{n} \right\}$, the remarkably simple condition $\gamma _{1}+\cdots +\gamma _{n} = \lambda _{1}+\cdots +\lambda _{n}$ is necessary and sufficient for the existence and construction of a realizing matrix with diagonal entries $\Gamma .$ Conditions for more general lists of complex numbers are also given.

Realizing Suleĭmanova spectra via permutative matrices, II

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 (2016) [8] is extended to include normalized lists that satisfy the new sufficient condition.