Computing nonnegative rank factorizations

The existence of nonnegative generalized inverses in terms of nonnegative rank factorizations is considered. An algorithm is presented which computes a nonnegative rank factorization of a nonnegative matrix when a nonnegative 1-inverse exists.     

A note on nonnegative rank factorizations

A nonnegative weakly monotone matrix of rank r has a nonnegative rank factorization if and only if it possesses an rxr monomial submatrix.     

The higher rank numerical range of nonnegative matrices

In this article the rank-k numerical range ∧ _k (A) of an entrywise nonnegative matrix A is investigated. Extending the notion of elements of maximum modulus in ∧ _k (A), we examine their location on the complex plane. Further, an application of this theory to ∧ _k (L(λ)) of a Perron polynomial L(λ) is elaborated via its companion matrix C _L .     

On the nonnegative rank of Euclidean distance matrices

The Euclidean distance matrix for n distinct points in R^r is generically of rank r + 2. It is shown in this paper via a geometric argument that its nonnegative rank for the case r = 1 is generically n.     

Real rank versus nonnegative rank

We consider the set of m×n nonnegative real matrices and define the nonnegative rank of a matrix A to be the minimum k such that A=BC where B is m×k and C is k×n. Given that the real rank of A is j for some j, we give bounds on the nonnegative rank of A and A².     

Irreducible factorizations of rationalq-parameter matrices

This paper considers rational q-parameter matrices (i.e., matrices the entries of which are ratios of scalar polynomials in q variables) and extends the previous results of the authors. Bibliography: 8 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 248, 1998, pp. 147–164. Translated by V. N. Kublanovskaya.     

Pseudoprime factorizations of integer matrices

We give a method of factoring integer matrices in into components such that the factorization is not unique unless certain information is known. In Section 2, we introduce this method of factorization and provide theorems which establish its well-definedness. In Section 3, we construct a matrix in as a product of specific types of matrices and establish an algorithm for factoring the result uniquely given an amount of information.     

Parallel factorizations of polynomial matrices

Conditions are established under which suggested factorizations of polynomial matrices over a field are parallel to factorizations of their canonical diagonal forms. An existence criterion of these factorizations of polynomial matrices is indicated and a method of constructing them is suggested.Translated from Ukrayins'kyy Matematychnyy Zhurnal, Vol. 44, No. 9, pp. 1228–1233, September, 1992.     

Intrinsic products and factorizations of matrices

We say that the product of a row vector and a column vector is intrinsic if there is at most one nonzero product of corresponding coordinates. Analogously we speak about intrinsic product of two or more matrices, as well as about intrinsic factorizations of matrices. Since all entries of the intrinsic product are products of entries of the multiplied matrices, there is no addition. We present several examples, together with important applications. These applications include companion matrices and sign-nonsingular matrices.     

Algebraic eigenspaces of nonnegative matrices

The Perron-Frobenius theory for square, irreducible, nonnegative matrices is generalized by studying the structure of the algebraic eigenspace of an arbitrary square nonnegative matrix corresponding to its spectral radius. We give a constructive proof that this subspace is spanned by a set of semipositive vectors and give a combinatorial characterization of both the index of the spectral radius and dimension of the algebraic eigenspace corresponding to the spectral radius. This involves a detailed study of the standard block triangular representation of nonnegative matrices by giving special attention to those blocks on the diagonal having the same spectral radius as the original matrix. We also show that the algebraic eigenspace corresponding to the spectral radius contains a semipositive vector having the largest set of positive coordinates among all vectors in this subspace.     

Alternative gradient algorithms with applications to nonnegative matrix factorizations

Three nonnegative matrix factorization (NMF) algorithms are discussed and employed to three real-world applications. Based on the alternative gradient algorithm with the iteration steps being determined columnwisely without projection, and columnwisely and elementwisely with projections, three algorithms are developed respectively. Also, the computational costs and the convergence properties of the new algorithms are given. The numerical examples show the advantage of our algorithms over the multiplicative update algorithm proposed by Lee and Seung [11].     

The rank of sparse random matrices

We determine the asymptotic normalized rank of a random matrix $\mathit{A}$ over an arbitrary field with prescribed numbers of nonzero entries in each row and column. As an application we obtain a formula for the rate of low-density parity check codes. This formula vindicates a conjecture of Lelarge (2013). The proofs are based on coupling arguments and a novel random perturbation, applicable to any matrix, that diminishes the number of short linear relations.     

On powers of nonnegative matrices

We investigate the structure of powers of nonnegative matrices A, and in particular characterize those A for which some power is (essentially) triangular. We also show how the number of irreducible components of A can be determined from its powers.     

Intervals of totally nonnegative matrices

Totally nonnegative matrices, i.e., matrices having all their minors nonnegative, and matrix intervals with respect to the checkerboard ordering are considered. It is proven that if the two bound matrices of such a matrix interval are nonsingular and totally nonnegative (and in addition all their zero minors are identical) then all matrices from this interval are also nonsingular and totally nonnegative (with identical zero minors).     

On nonnegative factorization of matrices

It is shown that a sufficient condition for a nonnegative real symmetric matrix to be completely positive is that the matrix is diagonally dominant.     

Generalized Leibniz functional matrices and factorizations of some well-known matrices

In this paper, we introduce the generalized Leibniz functional matrices and study some algebraic properties of such matrices. To demonstrate applications of these properties, we derive several novel factorization forms of some well-known matrices, such as the complete symmetric polynomial matrix and the elementary symmetric polynomial matrix. In addition, by applying factorizations of the generalized Leibniz functional matrices, we redevelop the known results of factorizations of Stirling matrices of the first and second kind and the generalized Pascal matrix.