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Applications 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.

The nonnegative inverse eigenvalue problem is the problem of finding necessary and sufficient conditions for the existence of an entrywise nonnegative matrix A with prescribed spectrum. This problem remains open for . If the matrix A is required to be normal, the problem will be called the normal nonnegative inverse eigenvalue problem (NNIEP). Sufficient conditions for a list of complex numbers to be the spectrum of a normal nonnegative matrix were obtained by Xu [Linear Multilinear Algebra. 1993;34:353–364]. In this paper, we give a normal version of a rank-r perturbation result due to Rado and published by Perfect [Duke Math. J. 1955;22:305–311], which allow us to obtain new sufficient conditions for the NNIEP to have a solution. These new conditions significantly improve Xu’s conditions. We also apply our results to construct nonnegative matrices with arbitrarily prescribed elementary divisors.

The goal of this paper is to study the switched stochastic control problem of discrete-time linear systems with multiplicative noises. We consider both the quadratic and the H∞ criteria for the performance evaluation. Initially we present a sufficient condition based on some Lyapunov–Metzler inequalities to guarantee the stochastic stability of the switching system. Moreover, we derive a sufficient condition for obtaining a Metzler matrix that will satisfy the Lyapunov–Metzler inequalities by directly solving a set of linear matrix inequalities, and not bilinear matrix inequalities as usual in the literature of switched systems. We believe that this result is an interesting contribution on its own. In the sequel we present sufficient conditions, again based on Lyapunov–Metzler inequalities, to obtain the state feedback gains and the switching rule so that the closed loop system is stochastically stable and the quadratic and H∞ performance costs are bounded above by a constant value. These results are illu...

In statistical practice, rectangular tables of numeric data are commonplace, and are often analyzed using dimension-reduction methods like the singular value decomposition and its close cousin, principal component analysis (PCA). This analysis produces score and loading matrices representing the rows and the columns of the original table and these matrices may be used for both prediction purposes and to gain structural understanding of the data. In some tables, the data entries are necessarily nonnegative (apart, perhaps, from some small random noise), and so the matrix factors meant to represent them should arguably also contain only nonnegative elements. This thinking, and the desire for parsimony, underlies such techniques as rotating factors in a search for “simple structure.” These attempts to transform score or loading matrices of mixed sign into nonnegative, parsimonious forms are, however, indirect and at best imperfect. The recent development of nonnegative matrix factorization, or NMF, is an att...

The basin of attraction to the origin in the space is obtained for the m dimensional system of difference equations of the formwhen coefficient is a Metzler matrix. Some information about the dynamics of the solutions outside of the basin is also given.

We present a nonnegative rank factorization of a nonnegative matrix A for the case in which one or both of A (1) A and AA (1) are nonnegative. This gives, in particular, a known result for characterizing nonnegative matrices when A † A or AA † is nonnegative. We applied this characterization to the derivation of known results based on the characterization of nonnegative monotone matrices.

For a square (0, 1, −1) sign pattern matrix S, denote the qualitative class of S by Q(S). In this article, we investigate the relationship between sign patterns and matrices that diagonalize an irreducible nonnegative matrix. We explicitly describe the sign patterns S such that every matrix in Q(S) diagonalizes some irreducible nonnegative matrix. Further, we characterize the sign patterns S such that some member of Q(S) diagonalizes an irreducible nonnegative matrix. Finally, we provide necessary and sufficient conditions for a multiset of real numbers to be realized as the spectrum of an irreducible nonnegative matrix M that is diagonalized by a matrix in the qualitative class of some S 2 NS sign pattern.

In this article we obtain a nonnegative rank factorization of nonnegative matrices A satisfying one or both of the following conditions: (i) AA † ⩽ 0 (ii) A † A ⩽ 0, thus providing a new set of conditions that guarantee the existence of a nonnegative least-squares solution of a linear system. Indeed, the characterization of such matrices improves some of the previous known conditions for the existence of a nonnegative least-squares solution of a linear system.

We present two versions of the same inequality, relating the maximal diagonal entry of a nonnegative matrix to its eigenvalues. We demonstrate a matrix factorization of a companion matrix, which leads to a solution of the nonnegative inverse eigenvalue problem (denoted the nniep) for 4×4 matrices of trace zero, and we give some sufficient conditions for a solution to the nniep for 5×5 matrices of trace zero. We also give a necessary condition on the eigenvalues of a 5×5 trace zero nonnegative matrix in lower Hessenberg form. Finally, we give a brief discussion of the nniep in restricted cases.

For a set σ with n complex numbers, some sufficient conditions are found for σ to be the spectrum of an n ×n normal (entrywise) nonnegative (positive) matrix. After proving a fundamental theorem and introducing the companion set σ′ of σ which consists of real numbers, we prove that if σ′ satisfies any known sufficient conditions for a real set to be the spectrum of a nonnegative matrix introduced by Suleimanova, Perfect, Salzmann and Kellogg respectively, then σ is the spectrum of an n×n normal nonnegative matrix.

In this paper we determine the nonnegativity structure of the principal components of an n× nnonnegative matrix Pin terms of the marked reduced graph , the minus M-matrix which can be associated with P. We then apply this result to consider various types of nonnegative bases for the Perron eigenspace of P which can be extracted from a certain nonnegative matrix which is a polynomial in P. We also obtain a characterization for the eigenprojection Pon the Perron eigenspace of Pto be, itself, a nonnegative matrix. Our results provide new proofs and extensions of results of Friedland and Schneider and of Hartwig, Neumann, and Rose.

Square nonnegative matrices with the property that the multiplicative monoid M(A) generated by the matrix A is finite are characterized in several ways. At first, the least general upper bound for the cardinality of M(A) is derived. Then it is shown that any square nonnegative matrix is cogredient to a lower triangular block form with the diagonal consisting of three blocks L, A 0, and M where L and M are strictly lower triangular, A 0 has no zero rows or columns, and M(A) is finite if and only if. M(A 0) is so. Several criteria for, M(A 0) to be finite are presented. One of the normal forms of A applies very well to the characterization of the nonnegative solutions of each of the matrix equations X k = 0, X k = 1, X k = X, and X k = X T where k > 1 is an integer. It also leads to a polynomial time algorithm for deciding whether or not M(A) is finite, if the entries of A are nonnegative rationals.

Let A be an n×n essentially nonnegative matrix and consider the linear differential system . We show that there exists a constant h(A)>0 such that the trajectory emanating from x0 reaches at a finite time t0 =t(x 0)⩾0 if and only if the sequence of points generated by a finite differences approximation from x0 , with time-step 0

Necessary conditions are developed for a system of the form Aλ=B with A≥0 and B≥ to have a least squares solution X which is nonnegative. Also it is shown that if a nonnegative matrix A has a nonnegative W-weighted { 1,3 -inverse for some nonnegative positive definite symmetric matrix W. then A has a nonnegative (1, 3) -inverse. As, a consequence of this, a short proof is obtained of a recent theorem of Jain and Egawa concerning nonnegative best approximate solutions (SIAM J. ALG. DISC. METH., 3 (1982), 197-213).

In a previous paper we proved that the diagonal elements of a totally nonnegative matrix are majorized by its eigenvalues. In this note we show that the majorization of a vector of nonnegative real numbers by another vector of nonnegative real numbers is not sufficient for the existence of a totally nonnegative matrix with diagonal elements taken from the entries of the majorized vector and eigenvalues taken from the entries of the majorizing vector.

We construct a symmetric matrix N having specified eigenvalues λ1 ⩾ λ2 ⩾ ⋯ ⩾ λ n , and give conditions on the {λ1) that N be nonnegative. The construction depends on the eigenvector x belonging to the maximal eigenvalue λ1.

Let be an (unknown) irreducible nonnegative matrix. Suppose only the following information on A is given: (1) its spectral radius ρ(A) (an example for such an information is in knowing that A is stochastic); (2) lower and upper bounds on each of the entries of A, namely two nonnegative matrices B and E such that B  A  B + E are given. The purpose of this paper is to bound the error of the dominant eigenvector of A. The technique used is as follows: the error vector is shown to satisfy a set of linear constraints. Then, a set of linear programming problems is solved to obtain bounds on the values for the entries of the error vector.

MP matrices are those real matrices which possess a nonnegative, nonsingular l-inverse. This paper characterizes the nonnegative MP matrices and hence, determines when a nonnegative matrix A has a convergent regular splitting M—Q which induces the linear stationary iterative scheme x k+1=M −1 Qxk +M −1 b to solve Ax=b.

It is shown that a nonnegative matrix A generates a cyclic group of order s if and only if it is the direct sum of cr of the form s being the l.c.m of their orders and each satisfying the equality being taken modulo cr . For s=2 this was shown by Harary and Minc.

The Perron-Frobenius Theorem states that a matrix with nonnegative entries has at least one nonnegative eigenvalue of maximal absolute value and a corresponding eigenvector with nonnegative components. In this paper we discuss generalizations of this celebrated theorem that locate an eigenvalue of maximal absolute value and the components of a corresponding eigenvector within a certain angle of the complex plane depending on the angle which contains the entries of the matrix. A complete description of the 2 × 2 case as well as partial results for the general case are given.

Let λ be any subset of {1,2, 3, 4} containing 1 and consider the matrix equations (1) and AXA=A, (2) XAX=X, (3) AX=(AX) T (4) XA=(XA) T where A is an m×n real matrix. By a λ-inverse of A is meant a solution of the equations (i)ieλ. A result on binary relations is given and is used to obtain a characterization of all nonnegative matrices A having a nonnegative λ-inverse, for all possible λ.

A "measure of irreducibility" of a nonnegative matrix is introduced and applied to obtaining an estimate from below for the distance of the Perron eigenvalue from any other eigenvalue of such a matrix.

Liao Xiao Xin (1987) gave necessary and sufficient conditions for the stability of a class of interval matrices. It can easily be shown that these results follow directly from a theorem relating stable Metzler matrices and quasidominant negative diagonal matrices by McKenzie (1966) which states (see also Siljak 1978) A Metzler matrix A is stable if and only if it is quasidominant negative diagonal. As the author considers Metzler matrices, Theorems 1,2 and 3 are obvious. Also, a theorem by Fiedler and Ptak (1962) concerning a pair of Minkowski matrices (see also Siljak 1978) leads to the same results.