Nonnegative Inverse Research Articles

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.

An Algorithm for Solving Quadratic Programming Problems with an M-matrix

In this study, we propose an approach for solving a quadraticprogramming problem with an M-matrix and simple constraints (QPs). It isbased on the algorithms of Luk-Pagano and Stachurski. These methods usethe fact that an M-matrix possesses a nonnegative inverse which allows tohave a sequence of feasible points monotonically increasing. Introducing theconcept of support for an objective function developed by Gabasov et al., ourapproach leads to a more general condition which allows to have an initialfeasible solution, related to a coordinator support and close to the optimalsolution. The programming under MATLAB of our method and that of Lukand Pagano has allowed us to make a comparison between them, with anillustration on two numerical examples.

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.

On the monotone convergence of Jacobi–Newton method for mildly nonlinear systems

For mildly nonlinear systems, involving concave or convex diagonal nonlinearities, semi-global monotone convergence of Newton’s method is guaranteed provided that the Jacobian of the system has a nonnegative inverse. However, regardless of this convergence result, the efficiency of Newton’s method becomes poor for stiff nonlinearities. We propose a nonlinear preconditioning procedure inspired by the Jacobi method and resulting in a new system of equations, which can be solved by Newton’s method much more efficiently. The obtained preconditioned method is shown to be globally convergent.

Nonnegative Inverse Elementary Divisors Problem for Lists with Nonnegative Real Parts

In this paper, sufficient conditions for the existence and construction of nonnegative matrices with prescribed elementary divisors for a list of complex numbers with nonnegative real part are obtained, and the corresponding nonnegative matrices are constructed. In addition, results of how to perturb complex eigenvalues of a nonnegative matrix while keeping its elementary divisors and its nonnegativity are derived.

The role of certain Brauer and Rado results in the nonnegative inverse spectral problems

It is said that a list $\Lambda =\{\lambda _{1},\ldots ,\lambda _{n}\}$ of complex numbers is realizable, if it is the spectrum of a nonnegative matrix $A$. It is said that $\Lambda $ is universally realizable if it is realizable for each possible Jordan canonical form allowed by $\Lambda$. This work does not contain new results. As its title says, its goal is to show and emphasize the relevance and importance of certain results, by Brauer and Rado, in the study of nonnegative inverse spectral problems. It is shown that virtually all known results, which give sufficient conditions for $\Lambda$ to be realizable or universally realizable, can be obtained from results by Brauer and Rado. Moreover, from these results, a realizing matrix may always be constructed.

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.

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.

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.

Block matrices and Guo's index for block circulant matrices with circulant blocks

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 λ0 of a realizable list is the minimum spectral radius such that the list (up to the initial spectral radius) together with λ0 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.

When are nonnegative matrices product of nonnegative idempotent matrices?

ABSTRACTApplications of nonnegative matrices have been of immense interest to both social and physical scientists, particularly to economists and statisticians. This paper considers the question as to when a nonnegative singular matrix can be decomposed as a product of nonnegative idempotents analogous to the well-known result for any arbitrary matrix. It is shown that (i) all singular nonnegative matrices of rank , (ii) all singular nonnegative matrices that have a nonnegative von Neumann inverse, (iii) (0-1) nonnegative definite matrices and (iv) periodic matrices have the property that they decompose into a product of nonnegative idempotents. An example is given, in general, showing that this need not be true for singular matrices of rank three or higher, including stochastic and symmetric matrices. Besides computational techniques, a recent result that a singular nonnegative quasi-permutation matrix is a product of nonnegative idempotents plays a key role in the proofs of the results.

Particle sizing by the Fraunhofer diffraction method based on an approximate non-negatively constrained Chin-Shifrin algorithm

In the Fraunhofer diffraction particle sizing technique there are relative errors between the particle size distribution (PSD) of the sample and the PSD recovered because there is random noise in the collected diffracted light intensity data and this makes data inversion to recover the PSD problematic. To reduce the noise sensitivity of the Chin-Shifrin inversion algorithm, an improved Chin-Shifrin algorithm is proposed that uses an approximate function to fit the integral kernel function together with a constraint to ensure non-negative inversion results. Furthermore, the tuning factor α of the approximate function is determined by analyzing the relationship between average particle size and the maximum value of the derivative of the diffracted light intensity. The PSDs of standard materials recovered by the Chin-Shifrin algorithm and its improved version are compared. Experimental data show that the improved Chin-Shifrin algorithm can effectively suppress random noise. The relative error of the recovered PSD is <5% and the relative standard deviation is no >10%.

A note on the real nonnegative inverse eigenvalue problem

The Real Nonnegative Inverse Eigenvalue Problem (RNIEP) asks when is a list \[ \sigma=(\lambda_1, \lambda_2,\ldots,\lambda_n)\] consisting of real numbers the spectrum of an $n \times n$ nonnegative matrix $A$. In that case, $\sigma$ is said to be realizable and $A$ is a realizing matrix. In a recent paper dealing with RNIEP, P.~Paparella considered cases of realizable spectra where a realizing matrix can be taken to have a special form, more precisely such that the entries of each row are obtained by permuting the entries of the first row. A matrix of this form is called permutative. Paparella raised the question whether any realizable list $\sigma$ can be realized by a permutative matrix or a direct sum of permutative matrices. In this paper, it is shown that in general the answer is no.

The odd case of Rota's bases conjecture

The paper links four conjectures:(1)(Rota's bases conjecture): For any system A=(A1,…,An) of non-singular real valued matrices the multiset of all columns of matrices in A can be decomposed into n independent systems of representatives of A.(2)(Alon–Tarsi): For even n, the number of even n×n Latin squares differs from the number of odd n×n Latin squares.(3)(Stones–Wanless, Kotlar): For all n, the number of even n×n Latin squares with the identity permutation as first row and first column differs from the number of odd n×n Latin squares of this type.(4)(Aharoni–Berger): Let M and N be two matroids on the same vertex set, and let A1,…,An be sets of size n+1 belonging to M∩N. Then there exists a set belonging to M∩N meeting all Ai. Huang and Rota [8] and independently Onn [11] proved that for any n (2) implies (1). We prove equivalence between (2) and (3). Using this, and a special case of (4), we prove the Huang–Rota–Onn theorem for n odd and a restricted class of input matrices: assuming the Alon–Tarsi conjecture for n−1, Rota's conjecture is true for any system of non-singular real valued matrices where one of them is non-negative and the remaining have non-negative inverses.

非负迭代截断奇异值纳米颗粒粒度分布反演算法

非负迭代截断奇异值纳米颗粒粒度分布反演算法

Inverse Nonnegativity of Tridiagonal &amp;lt;i&amp;gt;M&amp;lt;/i&amp;gt;-Matrices under Diagonal Element-Wise Perturbation

One of the most important properties of M-matrices is element-wise non-negative of its inverse. In this paper, we consider element-wise perturbations of tridiagonal M-matrices and obtain bounds on the perturbations so that the non-negative inverse persists. The largest interval is given by which the diagonal entries of the inverse of tridiagonal M-matrices can be perturbed without losing the property of total nonnegativity. A numerical example is given to illustrate our findings.

On Sufficient Conditions for the RNIEP and their Margins of Realizability

The Real Nonnegative Inverse Eigenvalue Problem (RNIEP) is that of characterizing all possible real spectra of nonnegative matrices. In this work we list some inclusion relations between several sufficient conditions and we study the negativity and the realizability margin of a spectrum with respect to these conditions.

Constructing new realisable lists from old in the NIEP

Given a list of complex numbers σ:=(λ1,λ2,…,λn), 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 characterising all realisable lists.Given a realisable list (ρ,λ2,λ3,…,λm), where ρ is the Perron eigenvalue and λ2 is real, we find families of lists (μ1,μ2,…,μn), for which(μ1,μ2,…,μn,λ3,λ4,…,λm) is realisable. In addition, given a realisable list(ρ,α+iβ,α−iβ,λ4,λ5,…,λm), where ρ is the Perron eigenvalue and α and β are real, we find families of lists (μ1,μ2,μ3,μ4), for which (μ1,μ2,μ3,μ4,λ4,λ5,…,λm) is realisable.

动态光散射混合非负截断奇异值反演

动态光散射混合非负截断奇异值反演

The inverse positivity of perturbed tridiagonal M-matrices

A well-known property of an M-matrix M is that the inverse is element-wise non-negative, which we write as M-1⩾0. In this paper, we consider element-wise perturbations of non-symmetric tridiagonal M-matrices and obtain sufficient bounds on the perturbations so that the non-negative inverse persists. These bounds improve the bounds recently given by Kennedy and Haynes [Inverse positivity of perturbed tridiagonal M-matrices, Linear Algebra Appl. 430 (2009) 2312–2323]. In particular, when perturbing the second diagonals (elements (l,l+2) and (l,l-2)) of M, these sufficient bounds are shown to be the actual maximum allowable perturbations. Numerical examples are given to demonstrate the effectiveness of our estimates.