Related papers: Exponents of primitive symmetric companion matrice…

A nonnegative matrix $A$ is said to be {\it primitive} if for some positive integer $m$, entries in $A^m$ are positive, notationally represented as $A^m>0.$ The smallest such $m$ is called the {\it exponent} of $A$, denoted $exp(A).$ For…

A set of nonnegative matrices $\mathcal{M}=\{M_1, M_2, \ldots, M_k\}$ is called primitive if there exist indices $i_1, i_2, \ldots, i_m$ such that $M_{i_1} M_{i_2} \ldots M_{i_m}$ is positive (i.e. has all its entries $>0$). The length of…

An alternating sign matrix is a square matrix satisfying (i) all entries are equal to 1, -1 or 0; (ii) every row and column has sum 1; (iii) in every row and column the non-zero entries alternate in sign. The 8-element group of symmetries…

We prove that, over a field $\mathbb{F}$ of odd characteristic $p$, a companion matrix $C$ is the sum of $E$ and $N$, with $E$ $p$-potent (i.e. $E^p = E$,) and $N$ nilpotent, if and only if the trace of $C$ is an integer multiple of unity…

The $girth$ of a primitive Boolean matrix is defined to be the $girth$ of its associated digraph. In this paper, among all primitive Boolean matrices of order $n$, the primitive exponents of those of girth $g$ are considered. For the…

The goal of this paper is to study primitive groups that are contained in the union of maximal (in the symmetric group) imprimitive groups. The study of types of permutations that appear inside primitive groups goes back to the origins of…

We describe a class of matrices whose determinants are trivial to compute. A nice example of such a matrix is given by considering the symmetric matrix with entries {i+j choose i} (mod 2) in {0,1}, 0 <= i,j < n the binomial coefficients…

Orthogonal matrices which are linear combinations of permutation matrices have attracted enormous attention in quantum information and computation. In this paper, we provide a complete parametric characterization of all complex, real and…

In this paper we describe some properties of companion matrices and demonstrate some special patterns that arise when a Toeplitz or a Hankel matrix is multiplied by a related companion matrix. We present a new condition, generalizing known…

In this paper we give a simple, short, and self-contained proof for a non-trivial upper bound on the probability that a random $\pm 1$ symmetric matrix is singular.

A complete set of primitive orthogonal idempotents plays an important role in the representation theory of an associative algebra. In this paper, we construct a complete set of primitive orthogonal idempotents for any finite Brandt…

We define tensors, corresponding to cubic polynomials, which have the same exponent $\omega$ as the matrix multiplication tensor. In particular, we study the symmetrized matrix multiplication tensor $sM_n$ defined on an $n\times n$ matrix…

In this paper, we develop the theory of symmetric triads with multiplicities. First, we classify abstract symmetric triads with multiplicities. Second, we determine the symmetric triads with multiplicities corresponding to commutative…

We develop a procedure for determining whether a square complex matrix is unitarily equivalent to a complex symmetric (i.e., self-transpose) matrix. Our approach has several advantages over existing methods. We discuss these differences and…

This paper is a continuation of Almost Commutative Terwilliger Algebras of Group Association Schemes I: Classification [1]. In that paper, we found all groups G for which the Terwilliger algebra of the group association scheme, denoted T…

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…

A nonnegative matrix $A$ is called primitive if $A^k$ is positive for some integer $k>0$. A generalization of this concept to finite sets of matrices is as follows: a set of matrices $\mathcal M = \{A_1, A_2, \ldots, A_m \}$ is primitive if…

We present a formula for the trace of any symmetric power of a $n\times n$ matrix (with coefficients in a field) in terms of the ordinary powers of the matrix, an arbitrarily chosen linear function which vanishes on the identity matrix, and…

Over the finite field with two elements, we present a method for obtaining explicit expressions for the number of rank i matrices of the form A above B, where A is persymmetric (A matrix [a(i,j)] is persymmetric if a(i,j) = a(r,s) for i+j =…

This paper describes the homology of various simplicial complexes associated to set families from combinatorial number theory, including primitive sets, pairwise coprime sets, product-free sets, and coprime-free sets. We present a condition…