Nonnegative Eigenvalue 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.

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.

Energy estimate up to the boundary for stable solutions to semilinear elliptic problems

We obtain a universal energy estimate up to the boundary for stable solutions of semilinear equations with variable coefficients. Namely, we consider solutions to −Lu=f(u), where L is a linear uniformly elliptic operator and f is C1, such that the linearized equation −L−f′(u) has nonnegative principal eigenvalue. Our main result is an estimate for the L2+γ norm of the gradient of stable solutions vanishing on the flat part of a half-ball, for any nonnegative and nondecreasing f. This bound only requires the elliptic coefficients to be Lipschitz. As a consequence, our estimate continues to hold in general C1,1 domains if we further assume the nonlinearity f to be convex. This result is new even for the Laplacian, for which a C3 regularity assumption on the domain was needed.

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

Bounding the row sum arithmetic mean by Perron roots of row-permuted matrices

R+n×n denotes the set of n×n non-negative matrices. For A∈R+n×n let Ω(A) be the set of all matrices that can be formed by permuting the elements within each row of A. Formally:Ω(A)={B∈R+n×n:∀i∃ a permutation ϕis.t.bi,j=ai,ϕi(j)∀j}. For B∈Ω(A) let ρ(B) denote the spectral radius or largest non-negative eigenvalue of B. We show that the arithmetic mean of the row sums of A is bounded by the maximum and minimum spectral radius of the matrices in Ω(A). Formally, we show thatminB∈Ω(A)⁡ρ(B)≤1n∑i=1n∑j=1nai,j≤maxB∈Ω(A)⁡ρ(B). For positive A we obtain necessary and sufficient conditions for these inequalities to become an equality. We also give criteria which an irreducible matrix C should satisfy so that ρ(C)=minB∈Ω(A)⁡ρ(B) or ρ(C)=maxB∈Ω(A)⁡ρ(B). These criteria are used to derive algorithms for finding such C when all the entries of A are positive.

A Polarimetric Decomposition and Copula Quantile Regression Approach for Soil Moisture Estimation From Radarsat-2 Data Over Vegetated Areas

This paper proposes a novel framework for probabilistic estimation of surface soil moisture (SSM) based on polarimetric decomposition and copula quantile regression, mainly focusing on solving the low correlation between synthetic aperture radar (SAR) backscattering coefficients and SSM in corn-covered areas. Cloude-Pottier decomposition and adaptive non-negative eigenvalue decomposition can extract more polarization parameters, explaining the implicit information in polarization data from different theoretical levels. Polarization parameters and the backscattering coefficients for different polarizations constitute predictor variable parameters for estimating the SSM. The dimensionality of the predictor variable parameters is reduced by supervised principal component analysis (SPCA) to derive the first principal component. SPCA ensures a high correlation between the first principal component and the SSM. Finally, the Archimedes copula function simply and effectively constructs the nonlinear relationship between SSM and the first principal component to complete the quantile regression estimation of SSM. Results show that the root-mean-square error (RMSE) range of SSM estimation is 0.039-0.078 cm <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$^{3}$</tex-math></inline-formula> /cm <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$^{3}$</tex-math></inline-formula> and the correlation coefficient (R) is 0.401-0.761. In addition, copula quantile regression constructs an uncertainty range for the SSM estimate, which can be used to judge the reliability of the estimate.

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.

Investigation of Polarimetric Decomposition for Arctic Summer Sea Ice Classification Using Gaofen-3 Fully Polarimetric SAR Data

The aim of this article was to investigate the potential of polarimetric decomposition of Chinese Gaofen-3 (GF-3) C-band fully polarimetric synthetic aperture radar (PolSAR) data for Arctic sea ice classification during summer season. Five different polarimetric decomposition approaches, including the Cloude-Pottier decomposition (Cloude), the Freeman three-component decomposition (Freeman3), the Freeman three-component decomposition using the extended Bragg model (Freeman3X), the Yamaguchi three-component decomposition (Yamaguchi3), and the nonnegative eigenvalue decomposition (NNED) were analyzed using 35 scenes of GF-3 PolSAR data collected over the Fram Strait, Arctic from June 14–18, 2017. Polarimetric features extracted from these five methods were evaluated and utilized to train random forest classifiers to classify open water (calm water and rough water) and sea ice types (melted ice, unmelted ice, and deformed ice). The results show that NNED could ensure physically valid decomposed powers while the other three model-based decompositions had negative values. In terms of sea ice classification, NNED had the highest feature importance scores and achieved an overall accuracy and Kappa coefficient of about 86.18% and 0.82, respectively. Inclusion of radar incidence angle as a feature in the classifier could slightly improve the classification accuracy by about 3%. The influence of incidence angle on sea ice classification accuracy was also investigated and it was found that high incidence angles (39°–46°) were superior to low incidence angles (21°–27°) due to the overall higher accuracies.

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.

Existence of positive radial solutions for a semipositone elliptic equation

In this paper, we study the existence of positive radial solutions for a semipositone elliptic equation−Δu=λf(u)in Ω,u=0on ∂Ω, where Ω is a ball or an annulus in RN with N≥2, λ>0 is a parameter and f is a continuous function satisfying the semipositone condition f(0)<0. We give a weak and general sufficient condition on f for the existence of positive radial solutions for λ>0 large. We show that the linearized operator at each solution has the nonnegative first eigenvalue. These solutions are obtained as minimizers of a Lagrangian functional.