It is known that semi-magic square matrices form a 2-graded algebra or superalgebra with the even and odd subspaces under centre-point reflection symmetry as the two components. We show that other symmetries which have been studied for square matrices give rise to similar superalgebra structures, pointing to novel symmetry types in their complementary parts. In particular, this provides a unifying framework for the composite 'most perfect square' symmetry and the related class of 'reversible squares'; moreover, the semi-magic square algebra is identified as part of a 2-gradation of the general square matrix algebra. We derive explicit representation formulae for matrices of all symmetry types considered, which can be used to construct all such matrices.
Using the decomposition of semimagic squares into the associated and balanced symmetry types as a motivation, we introduce an equivalent representation in terms of blockstructured matrices. This block representation provides a way of constructing such matrices with further symmetries and of studying their algebraic behaviour, significantly advancing and contributing to the understanding of these symmetry properties. In addition to studying classical attributes, such as dihedral equivalence and the spectral properties of these matrices, we show that the inherent structure of the block representation facilitates the definition of low-rank semimagic square matrices. This is achieved by means of tensor product blocks. Furthermore, we study the rank and eigenvector decomposition of these matrices, enabling the construction of a corresponding two-sided eigenvector matrix in rational terms of their entries. The paper concludes with the derivation of a correspondence between the tensor product block representations and quadratic form expressions of Gaussian type. Block Representations and Dihedral SymmetriesDefinition. An n × n matrix M ∈ R n×n is called semimagic square of weight w if each of its rows and columns sums to nw. If, in addition, each of its two diagonals also adds up to nw, it is called a magic square.The following two centre-point symmetry types of semimagic square matrices are of interest.Definition. Let M be an n × n semimagic square matrix of weight w. (a) The matrix M is called associated if each entry and its mirror entry w.r.t. the centre of the matrix add to 2w, i.e. if M ij + M n+1−i,n+1−j = 2w (i, j ∈ {1, . . . , n}). (b) The matrix M is called balanced if each entry is equal to its mirror entry w.r.t. the centre of the matrix, i.e. M ij − M n+1−i,n+1−j = 0 (i, j ∈ {1, . . . , n}).
The jth divisor function d j , which counts the ordered factorisations of a positive integer into j positive integer factors, is a very well-known arithmetic function; in particular, d 2 (n) gives the number of divisors of n. However, the jth non-trivial divisor function c j , which counts the ordered proper factorisations of a positive integer into j factors, each of which is greater than or equal to 2, is rather less well-studied. We also consider associated divisor functions c (r) j , whose definition is motivated by the sum-over divisors recurrence for d j . After reviewing properties of d j , we study analogous properties of c j and c (r) j , specifically regarding their Dirichlet series and generating functions, as well as representations in terms of binomial coefficient sums and hypergeometric series. We also express their ratios as binomial coefficient sums and hypergeometric series, and find explicit Dirichlet series and Euler products in some cases. As an illustrative application of the non-trivial and associated divisor functions, we show how they can be used to count principal reversible square matrices of the type considered by Ollerenshaw and Brée, and hence sum-and-distance systems of integers.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.