Given a matrix M (not necessarily nonnegative) and a factorization rank r, semi-nonnegative matrix factorization (semi-NMF) looks for a matrix U with r columns and a nonnegative matrix V with r rows such that UV is the best possible approximation of M according to some metric. In this paper, we study the properties of semi-NMF from which we develop exact and heuristic algorithms. Our contributi...

The entropy of irreducible stochastic matrix measures its mixing properties. We show that this quantity is subadditive and strongly subadditive. 1Instytut Matematyki Uniwersytet Jagielloński Reymonta 4, 30-059 Kraków, Poland mail: [email protected] 1 1 Preliminaries Let N ∈ N. We denote the simplex of probability vectors (finite probability distributions) by ∆N = { p ∈ R : pi ≥ 0 for i = 1,...

Let A = (aij)n×n be a nonnegative, symmetric, irreducible and invertible matrix. We prove the existence and uniqueness of radial solutions to the following Liouville system with singularity:  ∆ui + ∑n j=1 aij |x| βj ej = 0, R2, i = 1, ..., n ∫ R2 |x| βieui(x)dx <∞, i = 1, ..., n where β1, ..., βn are constants greater than −2. If all βis are negative we prove that all solutions are radial and...

in this paper we present a method for solving fuzzy linear systemsby two crisp linear systems. also necessary and sufficient conditions for existenceof solution are given. some numerical examples illustrate the efficiencyof the method.

A deflation technique is presented for an irreducible singular M -matrix Algebraic Riccati Equation (MARE). The technique improves the rate of convergence of a doubling algorithm, especially for an MARE in the critical case for which without deflation the doubling algorithm converges linearly and with deflation it converges quadratically. The deflation also improves the conditioning of the MARE...

In this paper, we have proposed some new upper bounds for the largest Z-eigenvalue of an irreducible weakly symmetric and nonnegative tensor, which improve the known upper bounds obtained in Chang et al. (Linear Algebra Appl 438:4166-4182, 2013), Song and Qi (SIAM J Matrix Anal Appl 34:1581-1595, 2013), He and Huang (Appl Math Lett 38:110-114, 2014), Li et al. (J Comput Anal Appl 483:182-199, 2...

The power index of a square Boolean matrix A is the least integer d such that A is a linear combination of previous nonnegative powers of A. We determine the maximum power indices for the class of n × n primitive symmetric Boolean matrices of trace zero, the class of n × n irreducible nonprimitive symmetric Boolean matrices, and the class of n×n reducible symmetric Boolean matrices of trace zer...