Related papers: Power-product matrix: nonsingularity, sparsity and…

It is shown by Karp reduction that deciding the singularity of $(2^n - 1) \times (2^n - 1)$ sparse circulant matrices (SC problem) is NP-complete. We can write them only implicitly, by indicating values of the $2 + n(n + 1)/2$ eventually…

We consider n-by-n circulant matrices having entries 0 and 1. Such matrices can be identified with sets of residues mod n, corresponding to the columns in which the top row contains an entry 1. Let A and B be two such matrices, and suppose…

Nonnegative matrix factorization (NMF) has become a very popular technique in machine learning because it automatically extracts meaningful features through a sparse and part-based representation. However, NMF has the drawback of being…

The explicit formulas for the sums of positive powers of the integers $s_i$ unrepresentable by the triple of integers $d_1,d_2,d_3\in {\mathbb N}, \gcd(d_1,d_2,d_3)=1$, are derived.

A nonnegative matrix A is said to be primitive if there exists a positive integer m such that entries in A^m are positive and smallest such m is called the exponent of A: Primitive matrices are useful in the study of finite Markov chains…

We construct a categorification of the modular data associated with every family of unipotent characters of the spetsial complex reflection group $G(d,1,n)$. The construction of the category follows the decomposition of the Fourier matrix…

The problem of determining whether a diagonally dominant matrix is singular or nonsingular is a classical topic in matrix theory. This paper develops necessary and sufficient conditions for the singularity or nonsingularity of diagonally…

We prove two universality results for random tensors of arbitrary rank D. We first prove that a random tensor whose entries are N^D independent, identically distributed, complex random variables converges in distribution in the large N…

We give a formula for the determinant of an $n\times n$ matrix with entries from a commutative ring with unit. The formula can be evaluated by a "straight-line program" performing only additions, subtractions and multiplications of ring…

Let R_{n,d} be the ring of invariants of d-tuples of n x n matrices under the simultaneous conjugation action of the general linear group. A minimal generating system and a homogeneous system of parameters for R_{3,3} are determined.…

We discuss $d=1, {\cal N}=2$ supersymmetric matrix models and exhibit the associated $d=2$ collective field theory in the limit of dense eigenvalues. From this field theory we construct, by the addition of several new fields, a $d=2$…

Let $X = (x_1,\dots,x_n)$ be a vector of distinct positive integers. The $n \times n$ matrix $S = S(X) := (\gcd(x_i,x_j))_{i,j=1}^n$, where $\gcd(x_i,x_j)$ denotes the greatest common divisor of $x_i$ and $x_j$, is called the greatest…

A unified approach to the construction of weighing matrices and certain symmetric designs is presented. Assuming the weight $p$ in a weighing matrix $W(n,p)$ is a prime power, it is shown that there is a…

For all positive powers of primes $p\geq 5$, we prove the existence of infinitely many linear congruences between the exponents of twisted Borcherds products arising from a suitable scalar-valued weight $1/2$ weakly holomorphic modular form…

Let \svec = (s_1,...,s_m) and \tvec = (t_1,...,t_n) be vectors of nonnegative integer-valued functions of m,n with equal sum S = sum_{i=1}^m s_i = sum_{j=1}^n t_j. Let M(\svec,\tvec) be the number of m*n matrices with nonnegative integer…

We provide exact and asymptotic formulae for the number of unrestricted, respectively indecomposable, $d$-dimensional matrices where the sum of all matrix entries with one coordinate fixed equals 2.

For any relatively prime integers $r$ and $s$, let $a_{r,s}(n)$ denote the number of $(r,s)$-regular partitions of a positive integer of $n$ into distinct parts. Prasad and Prasad (2018) proved many infinite families of congruences modulo 2…

Let $V$ be a vector space of rectangular $n\times k$ matrices annihilating the Cullis' determinant. We show that $\dim(V) \le (n-1)k$, extending Dieudonn{\'{e}}'s result on the dimension of vector spaces of square matrices annihilating the…

It is well known that the dominant eigenvalue of a real essentially nonnegative matrix is a convex function of its diagonal entries. This convexity is of practical importance in population biology, graph theory, demography, analytic…

A multidimensional nonnegative matrix is called polystochastic if the sum of its entries at each line is equal to $1$. The set of all polystochastic matrices of order $n$ and dimension $d$ is a convex polytope $\Omega_n^d$. In the present…