Class Of Nonnegative Matrices Research Articles

Computing Closest Stable Nonnegative Matrix

The problem of finding the closest stable matrix for a dynamical system has many applications. It is studied for both continuous and discrete-time systems and the corresponding optimization problems are formulated for various matrix norms. As a rule, nonconvexity of these formulations does not allow finding their global solutions. In this paper, we analyze positive discrete-time systems. They also suffer from nonconvexity of the stability region, and the problem in the Frobenius norm or in the Euclidean norm remains hard for them. However, it turns out that for certain polyhedral norms, the situation is much better. We show that for the distances measured in the max-norm, we can find an exact solution of the corresponding nonconvex projection problems in polynomial time. For the distance measured in the operator $\ell_{\infty}$-norm or $\ell_{1}$-norm, the exact solution is also efficiently found. To this end, we develop a modification of the recently introduced spectral simplex method. On the other hand, for all these three norms, we obtain exact descriptions of the region of stability around a given stable matrix. In the case of the max-norm, this can be seen as an extension onto the class of nonnegative matrices, the Kharitonov theorem, providing a stability criterion for polynomials with interval coefficients [V. L Kharitonov, Differ. Uravn., 14 (1978), pp. 2086--2088; K. Panneerselvam and R. Ayyagari, Internat. J. Control Sci. Engrg., 3 (2013), pp. 81--85]. For practical implementation of our technique, we developed a new method for approximating the maximal eigenvalue of a nonnegative matrix. It combines the local quadratic rate of convergence with polynomial-time global performance guarantees.

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 $n > 2$ real numbers.

Heuristics for exact nonnegative matrix factorization

The exact nonnegative matrix factorization (exact NMF) problem is the following: given an $m$-by-$n$ nonnegative matrix $X$ and a factorization rank $r$, find, if possible, an $m$-by-$r$ nonnegative matrix $W$ and an $r$-by-$n$ nonnegative matrix $H$ such that $X = WH$. In this paper, we propose two heuristics for exact NMF, one inspired from simulated annealing and the other from the greedy randomized adaptive search procedure. We show that these two heuristics are able to compute exact nonnegative factorizations for several classes of nonnegative matrices (namely, linear Euclidean distance matrices, slack matrices, unique-disjointness matrices, and randomly generated matrices) and as such demonstrate their superiority over standard multi-start strategies. We also consider a hybridization between these two heuristics that allows us to combine the advantages of both methods. Finally, we discuss the use of these heuristics to gain insight on the behavior of the nonnegative rank, i.e., the minimum factorization rank such that an exact NMF exists. In particular, we disprove a conjecture on the nonnegative rank of a Kronecker product, propose a new upper bound on the extension complexity of generic $n$-gons and conjecture the exact value of (i) the extension complexity of regular $n$-gons and (ii) the nonnegative rank of a submatrix of the slack matrix of the correlation polytope.

Gau–Wu numbers of nonnegative matrices

For any n-by-n matrix A, we consider the maximum number k=k(A) of orthonormal vectors xj∈Cn such that the scalar products 〈Axj,xj〉 lie on the boundary ∂W(A) of the numerical range W(A). This number is called the Gau–Wu number of the matrix A. If A is an n-by-n (n≥2) nonnegative matrix with the permutationally irreducible real part of the form[0A100⋱⋱Am−100],where m≥3 and the diagonal zeros are zero square matrices, then k(A) has an upper bound m−1. In addition, we also obtain necessary and sufficient conditions for k(A)=m−1 for such a matrix A. Another class of nonnegative matrices we study is the doubly stochastic ones. We prove that the value of k(A) is equal to 3 for any 3-by-3 doubly stochastic matrix A. For any 4-by-4 doubly stochastic matrix, we also determine its numerical range, which is then applied to find its Gau–Wu numbers. Furthermore, a lower bound of the Gau–Wu number k(A) is also found for a general n-by-n (n≥5) doubly stochastic matrix A via the possible shapes of W(A).

Nonnegative Matrices That Are Similar to Positive Matrices

We study the question in which nonnegative matrices are similar to positive matrices first raised by Borobia and Moro in 1997. We identify certain classes of nonnegative matrices that are not similar to positive matrices and solve the problem when an $n\times n$ nonnegative matrix has exactly $n$ or $n+1$ elements equal to zero.

Nonnegative group-monotone matrices and the minus partial order

Bapat et al. previously described a class of nonnegative matrices dominated by a nonnegative idempotent matrix under the minus partial order. In this paper, we improve upon that description by first presenting a more general result that gives the precise structure of nonnegative matrices dominated by a group-monotone matrix under the minus partial order. As a special case we derive the complete class of nonnegative matrices dominated by a nonnegative idempotent matrix that includes the class obtained by Bapat et al.

A New Class of Inverse M-Matrices of Tree-Like Type

Previous article Next article A New Class of Inverse M-Matrices of Tree-Like TypeServet Martínez, Jaime San Martín, and Xiao-Dong ZhangServet Martínez, Jaime San Martín, and Xiao-Dong Zhanghttps://doi.org/10.1137/S0895479801396816PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAboutAbstractIn this paper, we use weighted dyadic trees to introduce a new class of nonnegative matrices whose inverses are column diagonally dominant M-matrices.[1] Google Scholar[2] Claude Dellacherie, , Servet Martínez and , Jaime San Martín, Ultrametric matrices and induced Markov chains, Adv. in Appl. Math., 17 (1996), 169–183 97h:15025 CrossrefISIGoogle Scholar[3] Claude Dellacherie, , Servet Martínez and , Jaime San Martín, Description of the sub‐Markov kernel associated to generalized ultrametric matrices. An algorithmic approach, Linear Algebra Appl., 318 (2000), 1–21 10.1016/S0024-3795(00)00193-2 2001h:15014 CrossrefISIGoogle Scholar[4] Miroslav Fiedler, Some characterizations of symmetric inverse M‐matrices, Proceedings of the Sixth Conference of the International Linear Algebra Society (Chemnitz, 1996), Vol. 275/276, 1998, 179–187 10.1016/S0024-3795(97)10022-2 99h:15024 Google Scholar[5] Miroslav Fiedler, Special ultrametric matrices and graphs, SIAM J. Matrix Anal. Appl., 22 (2000), 106–113 10.1137/S0895479899350988 2001d:15022 LinkISIGoogle Scholar[6] M. Fiedler, , Charles Johnson and , T. Markham, Notes on inverse M‐matrices, Linear Algebra Appl., 91 (1987), 75–81 10.1016/0024-3795(87)90061-9 88m:15002 CrossrefISIGoogle Scholar[7] Charles Johnson, Inverse M‐matrices, Linear Algebra Appl., 47 (1982), 195–216 10.1016/0024-3795(82)90233-6 83m:15003 CrossrefISIGoogle Scholar[8] T. L. Markham, Nonnegative matrices whose inverses are M‐matrices, Proc. Amer. Math. 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Willoughby, The inverse M‐matrix problem, Linear Algebra and Appl., 18 (1977), 75–94 10.1016/0024-3795(77)90081-7 57:12561 CrossrefISIGoogle ScholarKeywordsnonnegative matrixinverse M-matrixweighted dyadic tree Previous article Next article FiguresRelatedReferencesCited ByDetails Matrix Positivity11 September 2020 Cross Ref Ultrametric MatricesInverse M-Matrices and Ultrametric Matrices | 13 September 2014 Cross Ref LU-Factorization Versus Wiener-Hopf Factorization for Markov ChainsActa Applicandae Mathematicae, Vol. 128, No. 1 | 14 February 2013 Cross Ref Inverse M-matrices, IILinear Algebra and its Applications, Vol. 435, No. 5 | 1 Sep 2011 Cross Ref On zero-pattern invariant properties of structured matricesLinear Algebra and its Applications, Vol. 435, No. 4 | 1 Aug 2011 Cross Ref On the inverse mean first passage matrix problem and the inverse M-matrix problemLinear Algebra and its Applications, Vol. 434, No. 7 | 1 Apr 2011 Cross Ref Characterizations of inverse M-matrices with special zero patternsLinear Algebra and its Applications, Vol. 433, No. 5 | 1 Oct 2010 Cross Ref Path Product Matrices and Eventually Inverse M‐matricesCharles R. Johnson and Ronald L. SmithSIAM Journal on Matrix Analysis and Applications, Vol. 29, No. 2 | 19 March 2007AbstractPDF (126 KB) Volume 24, Issue 4| 2003SIAM Journal on Matrix Analysis and Applications History Published online:31 July 2006 InformationCopyright © 2003 Society for Industrial and Applied MathematicsKeywordsnonnegative matrixinverse M-matrixweighted dyadic treeMSC codes15A0905C5015A57PDF Download Article & Publication DataArticle DOI:10.1137/S0895479801396816Article page range:pp. 1136-1148ISSN (print):0895-4798ISSN (online):1095-7162Publisher:Society for Industrial and Applied Mathematics

Path product matrices

A new class of nonnegative matrices, called the path product (PP) matrices, is introduced. Every inverse M-matrix is PP and this fact gives transparent proofs of a number of facts about inverse M-matrices that hold in a broader setting. For n≤ 3 (and not greater) the (strict) PP matrices are exactly the inverse M-matrices. Finally, the PP matrices are studied, a number of properties given, and the completion problem is solved for partial PP matrices. Unlike other properties inherited by principal submatrices, there is no graphtheoretic restriction on completability of partial PP matrices.

Approximability by Weighted Norms of the Structured and VolumetricSingular Values of a Class of Nonnegative Matrices

A known result about the spectral radius of an irreducible nonnegative matrix is extended to all nonnegative matrices. By means of this result, it is shown that the structured singular value and the volumetric singular value of a class of nonnegative matrices can be approximated with arbitrary accuracy by the matrix norm induced by a weighted $\ell_2$ vector norm and in the simplest case by a weighted $\ell_p$ vector norm for any $p$.

Exponents of classes of non-negative matrices

AbstractA square non-negative matrix is called primitive if all the elements of some power of this matrix are positive. The exponent of a primitive matrix is the minimum power satisfying this condition. The exponent of a class of primitive matrices is the minimum power such that all the matrices of the class to this power have positive elements only. In this paper, bounds for the exponents of primitive matrices and classes of matrices are obtained.

Maximum permanents on certain classes of nonnegative matrices

Let U R (α, β) denote the class of all square matrices with each entry equal to one of the nonnegative numbers α and β and with row sum vector R. We prove that the maximum value of the permanent of a matrix in the convex hull of U R (α, β) is achieved at a matrix in U R (α, β). One consequence of this result is an extension to matrices with entries between 0 and 1 of the Minc-Brégman upper bound for permanents of (0,1) matrices in terms of the integral row sums. We obtain other upper bounds for permanents of certain nonnegative matrices and give a nonprobabilistic proof of an inequality of Chang for the permanents of certain doubly stochastic matrices.

Convex spectral functions

In this paper we characterize all convex functionals defined on certain convex sets of hermitian matrices and which depend only on the eigenvalues of matrices. We extend these results to certain classes of non-negative matrices. This is done by formulating some new characterizations for the spectral radius of non-negative matrices, which are of independent interest.