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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.

A real matrix is called totally nonnegative if all of its minors are nonnegative. In this paper, the minors are determined from which the maximum allowable entry perturbation of a totally nonnegative matrix can be found, such that the perturbed matrix remains totally nonnegative. Also, the total nonnegativity of the first and second subdirect sum of two totally nonnegative matrices is considered.

Arising from the classification of the matrix-roots of a nonnegative imprimitive irreducible matrix, we present results concerning the Jordan chains of an $h$-cyclic matrix. We also present ancillary results applicable to nonnegative imprimitive irreducible matrices and demonstrate these results via examples., Comment: To appear in the special issue "The Legacy of Hans Schneider" in Linear Algebra and its Applications

The max-times algebra is the set R + of nonnegative reals with operations ⊕ : ( a , b ) → max ⁡ { a , b } and ⊙ : ( a , b ) → a b . We discuss the property of matrices to be squares of max-times or conventional nonnegative matrices. We prove that there exists a matrix having a conventional nonnegative square root but no max-times square root. Also, we present a set S of cardinality three for which there is a nonnegative matrix M with spectrum S, and every such M has both conventional and max-times square roots. These results answer two questions from the recent paper by Tam and Huang.

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.

We study the semialgebraic structure of $D_r$, the set of nonnegative tensors of nonnegative rank not more than $r$, and use the results to infer various properties of nonnegative tensor rank. We determine all nonnegative typical ranks for cubical nonnegative tensors and show that the direct sum conjecture is true for nonnegative tensor rank. We show that nonnegative, real, and complex ranks are all equal for a general nonnegative tensor of nonnegative rank strictly less than the complex generic rank. In addition, such nonnegative tensors always have unique nonnegative rank-$r$ decompositions if the real tensor space is $r$-identifiable. We determine conditions under which a best nonnegative rank-$r$ approximation has a unique nonnegative rank-$r$ decomposition: for $r \le 3$, this is always the case; for general $r$, this is the case when the best nonnegative rank-$r$ approximation does not lie on the boundary of $D_r$. Many of our general identifiability results also apply to real tensors and real symmetric tensors., 25 pages, to appear in SIMAX

The purpose of this note is two-fold: (1) to study when quasi-Euclidean rings, regular rings and regular separative rings have the property (∗) that each right (left) singular element is a product of idempotents, and (2) to consider the question: “when is a singular nonnegative square matrix a product of nonnegative idempotent matrices?” The importance of the class of quasi- Euclidean rings in connection with the property (∗) is given by the first three authors and T.Y. Lam [Journal of Algebra, 406:154–170, 2014], where it is shown that every singular matrix over a right and left quasi-Euclidean domain is a product of idempotents, generalizing the results of J. A Erdos [Glasgow Mathematical Journal, 8: 118–122, 1967] for matrices over fields and that of T. J. Laffey [Linear and Multilinear Algebra, 14:309–314, 1983] for matrices over commutative Euclidean domains. We have shown in this paper that quasi-Euclidean rings appear among many interesting classes of rings and hence they are in abundance. We analyze the properties of triangular matrix rings and upper triangular matrices with respect to the decomposition into product of idempotents and show, in particular, that nonnegative nilpotent matrices are products of nonnegative idempotent matrices. We study as to when each singular matrix is a product of idempotents in special classes of rings. Regarding the second question for nonnegative matrices, bounds are obtained for a rank one nonnegative matrix to be a product of two idempotent matrices. It is shown that every nonnegative matrix of rank one is a product of three nonnegative idempotent matrices. For matrices of higher orders, we show that some power of a group monotone matrix is a product of idempotent matrices.

Metzler matrices are algebraically characterized by having nonnegative off-diagonal entries and, from the dynamical point of view, define continuous-time positive systems. Our work studies polytopes of Metzler matrices and linear dynamics generated by such sets. The first part of the paper explores the algebraic properties of matrix polytopes, by focusing on the estimation of a right outer bound for all eigenvalues. The second part analyzes the dynamical properties of positive polytopic systems, by revealing connections between the estimated right outer bound and the evolution of trajectories (related to copositive Lyapunov functions and invariant sets).

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.

Tridiagonal matrices are considered which are totally nonnegative, i. e., all their mi- nors are nonnegative. The largest amount is given by which the single entries of such a matrix can be perturbed without losing the property of total nonnegativity.

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...

This paper studies the construction problems of five order nonnegative matrices from spectrum data. Let be a list of complex numbers with . Necessary and sufficient conditions for the existence of an entry-wise nonnegative 5×5 matrix with spectrum are presented.

Given any nonnegative matrix $A \in \mathbb{R}^{m \times n}$ , it is always possible to express A as the sum of a series of nonnegative rank-one matrices. Among the many possible representations of A, the number of terms that contributes the shortest nonnegative rank-one series representation is called the nonnegative rank of A. Computing the exact nonnegative rank and the corresponding factorization are known to be NP-hard. Even if the nonnegative rank is known a priori, no simple procedure exists presently that is able to perform the nonnegative factorization. Based on the Wedderburn rank reduction formula, this paper proposes a heuristic approach to tackle this difficult problem numerically. Starting with A, the idea is to recurrently extrat, whenever possible, a rank-one nonnegative portion from the previous matrix while keeping the residual nonnegative and lowering its rank by one. With a slight modification for symmetry, the method can equally be applied to another important class of completely positive matrices. No convergence can be guaranteed, but repeated restart might help alleviate the difficulty. Extensive numerical testing seems to suggest that the proposed algorithm might serve as a first-step numerical means for exploring the intriguing problem of nonnegative rank factorization.

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.

In this paper, for a given set of numbers with special conditions, we construct a nonnegative matrix such that the given set is its spectrum.

In Nonnegative Matrix Factorization (NMF), a nonnegative matrix is approximated by a product of lower-rank factorizing matrices. Most NMF methods assume that each factorizing matrix appears only once in the approximation, thus the approximation is linear in the factorizing matrices. We present a new class of approximative NMF methods, called Quadratic Nonnegative Matrix Factorization (QNMF), where some factorizing matrices occur twice in the approximation. We demonstrate QNMF solutions to four potential pattern recognition problems in graph partitioning, two-way clustering, estimating hidden Markov chains, and graph matching. We derive multiplicative algorithms that monotonically decrease the approximation error under a variety of measures. We also present extensions in which one of the factorizing matrices is constrained to be orthogonal or stochastic. Empirical studies show that for certain application scenarios, QNMF is more advantageous than other existing nonnegative matrix factorization methods.

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.

An algorithm of diagonal transformation for the Perron root of nonnegative matrices is proposed by Duan and Zhang [F. Duan, K. Zhang, An algorithm of diagonal transformation for Perron root of nonnegative irreducible matrices, Appl. Math. Comput. 175 (2006) 762–772]. This method can be used for all nonnegative irreducible matrices. In this paper, an improved algorithm which is based on this method is proposed. The new algorithm inherits all the above-mentioned advantages of the original algorithm and has higher efficiency. It is testified by numerical testing that the efficiency of the new algorithm is improved greatly.

We present in this paper a general formulation for nonnegative data factorization, called projective nonnegative graph embedding (PNGE), which 1) explicitly decomposes the data into two nonnegative components favoring the characteristics encoded by the so-called intrinsic and penalty graphs , respectively, and 2) explicitly describes how to transform each new testing sample into its low-dimensional nonnegative representation. In the past, such a nonnegative decomposition was often obtained for the training samples only, e.g., nonnegative matrix factorization (NMF) and its variants, nonnegative graph embedding (NGE) and its refined version multiplicative nonnegative graph embedding (MNGE). Those conventional approaches for out-of-sample extension either suffer from the high computational cost or violate the basic nonnegative assumption. In this work, PNGE offers a unified solution to out-of-sample extension problem, and the nonnegative coefficient vector of each datum is assumed to be projected from its original feature representation with a universal nonnegative transformation matrix. A convergency provable multiplicative nonnegative updating rule is then derived to learn the basis matrix and transformation matrix. Extensive experiments compared with the state-of-the-art algorithms on nonnegative data factorization demonstrate the algorithmic properties in convergency, sparsity, and classification power.

Let A be an n × n irreducible nonnegative matrix. We show that over the set Ω n of all n × n doubly stochastic matrices S, the multiplicative spectral radius ρ ( SA ) attains a minimum and a maximum at a permutation matrix. For the case when A is a symmetric nonnegative matrix, a by-product of our technique of proof yields a result allowing us to show that ρ ( S 1 A ) ⩾ ρ ( S 2 A ) , when S 1 and S 2 are two symmetric matrices such that both S 1 A and S 2 A are nonnegative matrices and S 1 - S 2 is a positive semidefinite matrix. This result has several corollaries. One corollary is that ρ ( S 1 A ) ⩾ ρ ( S 2 A ) , when S 1 = ( 1 / n ) J and S 2 = ( 1 / ( n - 1 ) ) ( J - I ) , where J is the matrix of all 1’s. A second corollary is a comparison theorem for weak regular splittings of two monotone matrices.

In this paper, we study the structured nonnegative matrix factorization problem: given a square, nonnegative matrix P , decompose it as P = VAV ⊤ with V and A nonnegative matrices and with the dimension of A as small as possible. We propose an iterative approach that minimizes the Kullback–Leibler divergence between P and VAV ⊤ subject to the nonnegativity constraints on A and V with the dimension of A given. The approximate structured decomposition P ≃ VAV ⊤ is closely related to the approximate symmetric decomposition P ≃ VV ⊤ . It is shown that the approach for finding an approximate structured decomposition can be adapted to solve the symmetric decomposition problem approximately. Finally, we apply the nonnegative decomposition VAV ⊤ to the hidden Markov realization problem and to the clustering of data vectors based on their distance matrix.

For a class of positive matrices A + K with a stable positive nominal part A and a structured positive perturbation part K, we address the problem of finding the largest set of admissible perturbations such that the global matrix remains stable. Theoretical bounds are derived and an algorithm for constructing this set is presented. As an example, this algorithm is applied to the regulation of water flow in open channels. Copyright (C) 2008 John Wiley & Sons. Ltd.

We study the concept of a nonnegative matrix inducing hierarchies and apply it to the nonnegative input–output matrix of the open Leontief model, leading to properties which are stronger than the feasibility of the model. We show some examples of these matrices. The characterization of nonnegative matrices inducing hierarchies for any permutation of indices leads to certain matrices of class M ( x ) .

Linear differential systems $\dot{x}(t)=Ax(t)$ ($A\in\mathbb{R}^{n\times n}$, $x_0=x(0)\in\mathbb{R}^n$, $t\geq0$) whose solutions become and remain nonnegative are studied. It is shown that the eigenvalue of $A$ furthest to the right must be real and must possess nonnegative right and left eigenvectors. Moreover, for some $a\geq0$, $A+aI$ must be eventually nonnegative, that is, its powers must become and remain entrywise nonnegative. Initial conditions $x_0$ that result in nonnegative states $x(t)$ in finite time are shown to form a convex cone that is related to the matrix exponential $e^{tA}$ and its eventual nonnegativity.

In this paper, we obtain some new bounds for Perron root of a nonnegative matrix, which are expressed by easily calculated function in element of matrix. These new results generalize and improve the bounds of G. Frobenius [1] and H. Minc [2], and also extend the known results by Liu [6].

Nonnegative matrix factorization (NMF) is a powerful technique for dimensionality reduction. Conventional NMF algorithms usually keep the matrices W and H nonnegative while iterating. However, to get the NMF of a matrix, it's unnecessary to force the temporary solutions in iterations nonnegative. In this paper, we propose a two-staged approach for NMF. At the relaxation stage, the nonnegative constraint of temporary solutions is relaxed and a real valued matrix factorization is generated. At the constraint stage, the real valued matrix factorization is transformed to a nonnegative matrix factorization by an invertible linear transformation. Based on this approach, we study on exact nonnegative matrix factorization when rank=2. We proved that, given two real valued matrices of rank=2, there exists an invertible linear transformation which can transform the real valued matrices to nonnegative matrices with their product stable. We propose an algorithm to find out the transformation. When rank is higher than 2, this kind of transformation may not exist. In the experiments, it's showed that this approach can reach a nonnegative matrix factorization with lower reconstruction error than conventional methods, and the technique for rank=2 exact NMF works well.

How quickly can we compute the nonnegative rank (r) of an m x n matrix? This problem ---- and the companion problem of finding a nonnegative matrix factorization with minimum inner-dimension ---- has a rich history, with applications in quantum mechanics, probability theory, data analysis, communication complexity and polyhedral combinatorics. Here we will survey the recent progress on this question, that has essentially resolved its worst-case complexity.

A nonnegative sign pattern matrix is a matrix whose entries are from the set {+, 0}. A nonnegative sign pattern matrix can also be viewed as a Boolean matrix, by replacing each + entry with 1. In this paper, some interesting connections between nonnegative sign pattern matrices and Boolean matrices are investigated. In particular, the relations between the minimum rank and the Boolean row (or column) rank are explored; the idempotent Boolean matrices that allow idempotence are identified; and the nonnegative sign patterns that allow various types of nonnegative (or positive) generalized inverses are characterized.

Let V∈ R m,n be a nonnegative matrix. The nonnegative matrix factorization (NNMF) problem consists of finding nonnegative matrix factors W∈ R m,r and H ∈ R r , n such that V ≈ WH . Lee and Seung proposed two algorithms, one of which finds nonnegative W and H such that ∥ V − WH ∥ F is minimized. After examining the case in which r =1 about which a complete characterization of the solution is possible, we consider the case in which m = n and V is symmetric. We focus on questions concerning when the best approximate factorization results in the product WH being symmetric and on cases in which the best approximation cannot be a symmetric matrix . Finally, we show that the class of positive semidefinite symmetric nonnegative matrices V generated via a Soules basis admit for every 1⩽ r ⩽rank( V ), a nonnegative factorization WH which coincides with the best approximation in the Frobenius norm to V in R n,n of rank not exceeding r . An example of applications in which NNMF factorizations for nonnegative symmetric matrices V arise is video and other media summarization technology where V is obtained from a distance matrix.

In this paper we obtain a decomposition of nonnegative matrices A such that AA# ≥ 0. We then use this characterization to obtain the previous results known for nonnegative matrices A with A 3 ≥ 0. We also consider nonnegative matrices A with A - A 2 ≥ 0.

In this paper we give necessary and sufficient conditions for a matrix in Jordan canonical form to be similar to an eventually nonnegative matrix whose irreducible diagonal blocks satisfy the conditions identified by Zaslavsky and Tam, and whose subdiagonal blocks (with respect to its Frobenius normal form) are nonnegative. These matrices are referred to as seminonnegative matrices, and we show that they exhibit many of the same combinatorial spectral properties as nonnegative matrices. This paper extends the work on Jordan forms of irreducible eventually nonnegative matrices to the reducible case.

We consider the following inverse spectrum problem for nonnegative matrices: given a set of real numbers σ={λ1,λ2,…,λn}, find necessary and sufficient conditions for the existence of an n×n nonnegative matrix A with spectrum σ. In particular, by the use of a relevant theorem of Brauer we obtain new simple sufficient conditions for the problem to have a solution. Moreover, we can always construct a solution matrix, which is nonnegative generalized stochastic.

The following inverse spectrum problem for nonnegative matrices is considered: given a set of complex numbers σ ={ λ 1 , λ 2 ,…, λ n }, find necessary and sufficient conditions for the existence of an n × n nonnegative matrix A with spectrum σ . Our work is motivated by a relevant theoretical result of Guo Wuwen [Linear Algebra Appl. 266 (1997) 261, Theorem 2.1]: there exists a real parameter λ 0 ⩾max 2⩽ j ⩽ n | λ j | such that the problem has a solution if and only if λ 1 ⩾ λ 0 . In particular, we discuss how to compute λ 0 and the solution matrix A for certain class of matrices. A sufficient condition for the problem to have a solution is also derived.

Let A be an n×n nonnegative real matrix. We know that A is Γ-monotone if it has a nonnegative Γ-inverse. Moreover if A has an appropriate generalized inverse, then we can obtain nonnegative solutions for the linear system Ax=b, where b⩾0.In this paper, using properties about nonnegatives matrices, we present sufficient conditions in order that the Leontief price model(I+F)p(k+1)=[1+r(k)]Fp(k)+Gp0(k+1)a0,where F=(I−AT)−1BT and G=(I−AT)−1, have nonnegative solutions.

The notions of primitivity and exponent of a square nonnegative matrix A are classical: A is primitive provided there is a nonnegative integer k such that A k is entrywise positive and in the case A is primitive the exponent of A is the smallest such k. Fornasini and Valcher have extended the notion of primitivity to pairs (A, B) of square nonnegative matrices of the same order. The pair (A, B) is primitive provided there exist nonnegative integers h and k such that the sum of all products formed by words consisting of hA ’s and kB ’s is entrywise positive. This paper defines the exponent of a nonnegative matrix pair to be the smallest value of h + k over all such h and k. It is then shown that the largest exponent of a primitive pair of n by n nonnegative matrices lies in the interval [(n 3 − 5n 2 )/2 ,( 3n 3 + 2n 2 − 2n)/2]. In addition, the exponent of a pair of nonnegative matrices is related to properties of an associated twodimensional dynamical system. © 2001 Elsevier Science Inc. All rights reserved.

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.

The most widely studied matrix function is the matrix exponential, mainly because of its key role in the solution of differential equations. In this work we derive new alternatives for efficient computation of the matrix cosine. We focus especially on the two classes of Normal and Nonnegative matrices for which we study the interval of application for rational [7/7] L∞ approximation compared to the corresponding [7/7] Pade approximant when we implement the scaling and squaring Algorinthm [2].

We present a lower and an upper bound for the second smallest eigenvalue of Laplacian matrices in terms of the averaged minimal cut of weighted graphs. This is used to obtain an upper bound for the real parts of the non-maximal eigenvalues of irreducible nonnegative matrices. The result can be applied to Markov chains.

The upper bound of maximal entries in the principal eigenvector of a simple undirected connected graph is investigated in [Linear Algebra Appl. 310 (2000) 129]. We further investigate the maximal entry ymaxp in the principle eigenvector of symmetric nonnegative matrix with zero trace and obtain both sharp upper and lower bounds on the ymaxp. Particularly, this result answers the open question given in the above-mentioned reference

The spectral radius r(A) of matrix A is the maximum modulus of the Eigen values. In this paper, the studies about the lower and upper bounds for the spectral radius and the lower bounds for the minimum eigen value of appositive and nonnegative matrices are investigate.The matrix norm, the spectral radius norm,and the column (row) sums of nonnegative and positive matrices are widely used to establish some inequalities for matrices. Then several existing results are improved for these inequalities for nonnegative and positive matrix. Furthermore, the lower and upper bounds of the Perron roots for nonnegative matrices are examined, and some upper bounds are computed.

In this paper, we present a new variations of the popular nonnegative matrix factorization (NMF) approach to extend it to the data with negative values. When a NMF problem is formulated as μ ≈μμ, we try to develop a new method that only allows μ to contain nonnegative values, but allows both μ and μ to have both nonnegative and negative values. In this way, the original NMF is extended to be used for real value data matrix instead restricted to only negative value data matrix. To this end, we develops novel method to factorize the real value data matrix. The method is evaluated experimentally and the results showed its effectiveness.

A real n × n matrix A is said to be an M -matrix if there exists a nonnegative matrix B with maximal eigenvalue r such that A = α I — B , where α ≥ r . In this work, we present a sufficient condition for the existence of an M -matrix with prescribed complex spectrum.

In this note, we shall consider the sequence of consecutive powers in max algebra of a nonnegative matrix with each element less than or equal to 1. The notion of asymptotic period is defined to study the `limiting behavior' of this sequence. A simple and effective characterization for the asymptotic period of the sequence is given.

An m-by-n matrix A is called totally nonnegative if every minor of A is nonnegative. The Hadamard product of two matrices is simply their entry-wise product. This paper introduces the subclass of totally nonnegative matrices whose Hadamard product with any totally nonnegative matrix is again totally nonnegative. Many properties concerning this class are discussed including: a complete characterization for min {m,n} ; a characterization of the zero–nonzero patterns for which all totally nonnegative matrices lie in this class; and connections to Oppenheim's inequality.

Let A be an n X n nonnegative matrix. In this paper we consider the problems of maximizing the spectral radii of (i) A + ,X. and (ii) A + D, where X is a real n X n matrix whose Frobenius norm is restricted to be 1 and where D is as X but is further constrained to be a diagonal matrix. For both problems the maximums occur at nonnegative X and D, and we use tools of nonnegative matrices, most notably due to Levinger and Fiedler, as well as the Kuhn--Tucker criterion for constrained optimization, to find upper and lower bounds on the maximums, and, when A is additionally assumed to be irreducible, to characterize cases of equalities in these bounds. In the case of the first problem, when A is irreducible, we characterize a matrix which gives the global maximum. A matrix which yields a global maximum to the second problem is more complicated to characterize as, depending on A, the problem admits local maximums within the nonnegative diagonal matrices of Frobenius norm 1.

We consider the problem of nonnegative matrix factorization where the typical objective function is altered based on geometrical arguments. A noneuclidean geometry on positive real numbers is used to describe the nonnegative entries of a nonnegative matrix, influencing the factorization model. We design an optimization procedure from a differential geometric point of view for the newly proposed model. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)

The real nonegative eigenvalue problem is concerned with determining which n -tuples of real numbers are the spectrum of some n -by- n nonnegative matrix. Some questions are presented whose answer could help to characterize the boundary of this set, i.e., the set of those spectra of nonnegative matrices which are not the spectrum of any positive matrix.

WOS: 000071526500003, We present a sequence of progressively better upper bounds for the Perron root of a nonnegative matrix. Each element in the sequence is a function of the Perron root of the arithmetic symmetrization of a power of the matrix. The results complement those of Szyld based on the geometric symmetrization of a power of the matrix. (C) 1998 Elsevier Science Inc.