We consider the problem of characterizing nonnegativity of the Moore-Penrose inverse for matrix perturbations of the type A − XGY, when the Moore-Penrose inverse of A is nonnegative. Here, we say that a matrix B = (b ij ) is nonnegative and denote it by B ≥ 0 if b ij ≥ 0, ∀i, j. This problemwasmotivated by the results in [1], where the authors consider an M-matrix A and find sufficient conditio...

in this paper, we derive the necessary and sufficient conditions for the quaternion matrix equation xa=b to have the least-square bisymmetric solution and give the expression of such solution when the solvability conditions are met. futhermore, we consider the maximal and minimal inertias of the least-square bisymmetric solution to this equation. as applications, we derive sufficient and necess...

For the nonsymmetric algebraic Riccati equation for which the four coefficient matrices form an M -matrix, the solution of practical interest is often the minimal nonnegative solution. In this note we prove that the minimal nonnegative solution is positive when the M -matrix is irreducible.

Let A be a nonnegative idempotent matrix. It is shown that the Schur complement of a submatrix, using the Moore-Penrose inverse, is a nonnegative idempotent matrix if the submatrix has a positive diagonal. Similar results for the Schur complement of any submatrix of A are no longer true in general.

Nonnegative Matrix Factorization (NMF) has been widely used in machine learning and data mining. It aims to find two nonnegative matrices whose product can well approximate the nonnegative data matrix, which naturally lead to parts-based representation. In this paper, we present a local learning regularized nonnegative matrix factorization (LLNMF) for clustering. It imposes an additional constr...

Nonnegative matrix factorization (NMF) is a popular data analysis method, the objective of which is to approximate a matrix with all nonnegative components into the product of two nonnegative matrices. In this work, we describe a new simple and efficient algorithm for multi-factor nonnegative matrix factorization (mfNMF) problem that generalizes the original NMF problem to more than two factors...

A matrix A can be tested to determine whether it is eventually positive by ex1 amination of its Perron-Frobenius structure, i.e., by computing its eigenvalues and left and right 2 eigenvectors for the spectral radius ρ(A). No such “if and only if” test using Perron-Frobenius prop3 erties exists for eventually nonnegative matrices. The concept of a strongly eventually nonnegative 4 matrix was wa...

Nonnegative matrix factorization (NMF) seeks a decomposition of a nonnegative matrix XX0 into a product of two nonnegative factor matrices UX0 and VX0, such that a discrepancy between X and UV> is minimized. Assuming U 1⁄4 XW in the decomposition (for WX0), kernel NMF (KNMF) is easily derived in the framework of least squares optimization. In this paper we make use of KNMF to extract data, whic...

Matrix functions preserving several sets of generalized nonnegative matrices are characterized. These sets include PFn, the set of n×n real eventually positive matrices; and WPFn, the set of matrices A ∈ R such that A and its transpose have the Perron-Frobenius property. Necessary conditions and sufficient conditions for a matrix function to preserve the set of n× n real eventually nonnegative ...