A matrix is $k$-nonnegative if all its minors of size $k$ or less are nonnegative. We give a parametrized set generators and relations for the semigroup $n\times n$ invertible matrices in two special cases: when $k = n-1$ n-2$, restricted to unitriangular matrices. For these cases, we prove that can be partitioned into cells based on their factorizations generators, generalizing notion Bruhat f...

‎Let $G$ be a finite $p$-group of order $p^n$ and‎ ‎$|{mathcal M}(G)|=p^{frac{1}{2}n(n-1)-t(G)}$‎, ‎where ${mathcal M}(G)$‎ ‎is the Schur multiplier of $G$ and $t(G)$ is a nonnegative integer‎. ‎The classification of such groups $G$ is already known for $t(G)leq‎ ‎6$‎. ‎This paper extends the classification to $t(G)=7$.

The nonnegative rank of a nonnegative matrix is the smallest number of nonnegative rank-one matrices into which the matrix can be decomposed additively. Such decompositions are useful in diverse scientific disciplines. We obtain characterizations and bounds and show that the nonnegative rank can be computed exactly over the reals by a finite algorithm.

A square complex matrix A is eventually nonnegative if there exists a positive integer k0 such that for all k ≥ k0, A ≥ 0; A is strongly eventually nonnegative if it is eventually nonnegative and has an irreducible nonnegative power. It is proved that a collection of elementary Jordan blocks is a Frobenius Jordan multiset with cyclic index r if and only if it is the multiset of elementary Jorda...

In 1907, Oskar Perron showed that a positive square matrix has a unique largest positive eigenvalue with a positive eigenvector. This result was extended to irreducible nonnegative matrices by Geog Frobenius in 1912, and to irreducible nonnegative tensors and weakly irreducible nonnegative tensors recently. This result is a fundamental result in matrix theory and has found wide applications in ...

Normalized nonnegative models assign probability distributions to users and random variables to items; see [Stark, 2015]. Rating an item is regarded as sampling the random variable assigned to the item with respect to the distribution assigned to the user who rates the item. Models of that kind are highly expressive. For instance, using normalized nonnegative models we can understand users’ pre...

Factor widths of nonnegative integral positive semidefinite square matrices are investigated. The nonnegative factor width, the exact factor width and the binary factor width of such matrices are introduced. Some lower and upper bounds for these widths are obtained. Nonnegative symmetric (completely positive) matrices with some given nonnegative (binary) factor widths

Abstract. Any given nonnegative matrix A ∈ R can be expressed as the product A = UV for some nonnegative matrices U ∈ R and V ∈ R with k ≤ min{m, n}. The smallest k that makes this factorization possible 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 ...

Nonnegative nilpotent lower triangular completions of a nonnegative nilpotent matrix are studied. It is shown that for every natural number between the index of the matrix and its order, there exists a completion that has this number as its index. A similar result is obtained for the rank. However, unlike the case of complex completions of complex matrices, it is proved that for every nonincrea...