Related papers: Fooling-sets and rank in nonzero characteristic (e…

We consider the set $\mathcal M_n(\mathbb Z; H)$ of $n\times n$-matrices with integer elements of size at most $H$ and obtain upper bounds on the number of matrices from $\mathcal M_n(\mathbb Z; H)$, for which the characteristic polynomial…

Non-Negative Matrix Factorization (NMF) is a widely used dimension reduction method that factorizes a non-negative data matrix into two lower dimensional non-negative matrices: One is the basis or feature matrix which consists of the…

We study Helson matrices (also known as multiplicative Hankel matrices), i.e. infinite matrices of the form $M(\alpha) = \{\alpha(nm)\}_{n,m=1}^\infty$, where $\alpha$ is a sequence of complex numbers. Helson matrices are considered as…

The rank of an n x n matrix A is equal to the size of its largest square submatrix with a nonzero determinant, and it can be computed in O(n^2.37) time. Analogously, the size of the largest square submatrix with nonzero permanent is defined…

A novel lower bound is introduced for the full rank probability of random finite field matrices, where a number of elements with known location are identically zero, and remaining elements are chosen independently of each other, uniformly…

Most recent results in matrix completion assume that the matrix under consideration is low-rank or that the columns are in a union of low-rank subspaces. In real-world settings, however, the linear structure underlying these models is…

Suppose $\mathbb{F}$ is a field and let $\mathbf{a} := (a_1, a_2, \dotsc)$ be a sequence of non-zero elements in $\mathbb{F}$. For $\mathbf{a}_n := (a_1, \dotsc, a_n)$, we consider the family $\mathcal{M}_n(\mathbf{a})$ of $n \times n$…

We study the problem of low-rank matrix completion for symmetric matrices. The minimum rank of a completion of a generic partially specified symmetric matrix depends only on the location of the specified entries, and not their values, if…

In this note, we consider matrices similar to $X$-form matrices, which are the matrices for which only the diagonal and the anti-diagonal elements can be different from zero. First, we give a characterization of these matrices using the…

The exact nonnegative matrix factorization (exact NMF) problem is the following: given an $m$-by-$n$ nonnegative matrix $X$ and a factorization rank $r$, find, if possible, an $m$-by-$r$ nonnegative matrix $W$ and an $r$-by-$n$ nonnegative…

A completion of an m-by-n matrix A with entries in {0,1,*} is obtained by setting all *-entries to constants 0 or 1. A system of semi-linear equations over GF(2) has the form Mx=f(x), where M is a completion of A and f:{0,1}^n --> {0,1}^m…

A (q,k,t)-design matrix is an m x n matrix whose pattern of zeros/non-zeros satisfies the following design-like condition: each row has at most q non-zeros, each column has at least k non-zeros and the supports of every two columns…

In Linear Algebra over finite fields, a characteristic-dependent linear rank inequality is a linear inequality that holds by ranks of subspaces of a vector space over a finite field of determined characteristic, and does not in general hold…

Let $n,p,r$ be positive integers with $n \geq p\geq r$. A rank-$\overline{r}$ subset of $n$ by $p$ matrices (with entries in a field) is a subset in which every matrix has rank less than or equal to $r$. A classical theorem of Flanders…

A real square matrix $A$ of order $n \times n~ (n \geq 3)$ is called an $F_0$-matrix, if it is a $Z$-matrix (off-diagonal entries nonpositive), all of whose principal submatrices of orders at most $n-2$ are $M$-matrices while there is at…

Nonnegative matrix factorization (NMF) has become a widely used tool for the analysis of high-dimensional data as it automatically extracts sparse and meaningful features from a set of nonnegative data vectors. We first illustrate this…

The article concerns low-rank approximation of matrices generated by sampling a smooth function of two $m$-dimensional variables. We identify several misconceptions surrounding a claim that, for a specific class of analytic functions, such…

Let $r$ be a nonconstant noncommutative rational function in $m$ variables over an algebraically closed field $K$ of characteristic 0. We show that for $n$ large enough, there exists an $X\in M_n(K)^m$ such that $r(X)$ has $n$ distinct and…

Let $M=(E,{\cal I})$ be a matroid. A {\em $k$-truncation} of $M$ is a matroid {$M'=(E,{\cal I}')$} such that for any $A\subseteq E$, $A\in {\cal I}'$ if and only if $|A|\leq k$ and $A\in {\cal I}$. Given a linear representation of $M$ we…

We study completion of partial matrices with nonnegative entries to matrices of nonnegative rank at most $r$ for some $r \in \mathbb{N}$. Most of our results are for $r \leq 3$. We show that a partial matrix with nonnegative entries has a…