Related papers: Factorization of a prime matrix in even blocks
Motivated by Hirano-Tominaga's work \cite{HT} on rings for which every element is a sum of two idempotents and by de Seguins Pazzis's results \cite{de} on decomposing every matrix over a field of positive characteristic as a sum of…
Ordinary binary multiplication of natural numbers can be generalized in a non-trivial way to a ternary operation by considering discrete volumes of lattice hexagons. With this operation, a natural notion of `3-primality' -- primality with…
For a nonsingular integer matrix A, we study the growth of the order of A modulo N. We say that a matrix is exceptional if it is diagonalizable, and a power of the matrix has all eigenvalues equal to powers of a single rational integer, or…
This paper is concerned with factor left prime factorization problems for multivariate polynomial matrices without full row rank. We propose a necessary and sufficient condition for the existence of factor left prime factorizations of a…
This paper studies symmetric tensor decompositions. For symmetric tensors, there exist linear relations of recursive patterns among their entries. Such a relation can be represented by a polynomial, which is called a generating polynomial.…
It is widely known that the lower bound for the algorithmic complexity of square matrix multiplication resorts to at least $n^2$ arithmetic operations. The justification builds upon the following reasoning: given that there are $2 n^2$…
It is known that every complex square matrix with nonnegative determinant is the product of positive semi-definite matrices. There are characterizations of matrices that require two or five positive semi-definite matrices in the product.…
A matrix space of size $m\times n$ is a linear subspace of the linear space of $m\times n$ matrices over a field $\mathbb{F}$. The rank of a matrix space is defined as the maximal rank over matrices in this space. A matrix space…
The main results of this paper are twofold: the first one is a matrix theoretical result. We say that a matriz is superregular if all of its minors that are not trivially zero are nonzero. Given a a times b, a larger than or equal to b,…
We investigate the problem asking when any square matrix whose entries lie in a finite field of characteristic 2 is decomposable into the sum of a diagonalizable matrix and a nilpotent matrix with index of nilpotency at most 2 and, as a…
The decomposition matrix of a finite group in prime characteristic p records the multiplicities of its p-modular irreducible representations as composition factors of the reductions modulo p of its irreducible representations in…
If every element of a matrix group is similar to a permutation matrix, then it is called a permutation-like matrix group. References [4] and [5] showed that, if a permutation-like matrix group contains a maximal cycle of length equal to a…
We present an approach to decomposition and factor analysis of matrices with ordinal data. The matrix entries are grades to which objects represented by rows satisfy attributes represented by columns, e.g. grades to which an image is red, a…
Schemes for exact multiplication of small matrices have a large symmetry group. This group defines an equivalence relation on the set of multiplication schemes. There are algorithms to decide whether two schemes are equivalent. However, for…
We give conditions for when the tensor product of two positive maps between matrix algebras is a positive map. This happens when one map belongs to a symmetric mapping cone and the other to the dual cone. Necessary and sufficient conditions…
A recently-established necessary condition for polynomials that preserve the class of entrywise nonnegative matrices of a fixed order is shown to be necessary and sufficient for the class of nonnegative monomial matrices. Along the way, we…
In this note, we give a necessary and sufficient condition for a matrix A in M to be finitely G-determined, where M is the ring of 2 x 2 matrices whose entries are formal power series over an infinite field, and G is a group acting on M by…
In graph theory a partition of the vertex set of a graph is called equitable if for all pairs of cells all vertices in one cell have an equal number of neighbours in the other cell. Considering the implications for the adjacency matrix one…
The concepts of differentiation and integration for matrices are known. As far as each matrix is differentiable, it is not clear a priori whether a given matrix is integrable or not. Recently some progress was obtained for diagonalizable…
Totally equimodular matrices generalize totally unimodular matrices and arise in the context of box-total dual integral polyhedra. This work further explores the parallels between these two classes and introduces foundational building…