Block Representation and Spectral Properties of Constant Sum Matrices

Abstract

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}).

Integer matrix factorisations, superalgebras and the quadratic form obstruction

“…In the present paper, we revisit the question of integer matrix factorisation in the light of recent general results on matrix decompositions [7], [8]. We establish in Corollary 3.1 that the existence of integer solutions to a certain quadratic equation is a necessary condition for a matrix factorisation of the type M = N 2 or M = N T N (for symmetric positive definite M) to exist.…”

“…Multiplying the equation by 2 7 and setting x i = 16a i , w = 16w Z , we find that a necessary condition for the integer factorisation of the Wilson matrix is that there are integer solutions to the quadratic equation ( 5)…”

On the structure of additive systems of integers

“…, 6!− 1 exactly once. Rearranging the same factors in a different joint ordered factorisation, ((1, 5), (3, 3), (2, 2), (3, 2), (2, 2), (1, 3), (2, 2)), we obtain a different sum system with component sets of the same cardinalities n 1 , n 2 , n 3 , and the same target set, gives the corresponding sum system 1,48,49,96,97,144,145,192,193,240,241,288 7), (2,4), (5,2), (3,2), (4, 2), (2,5), (4,9), (3,3), (1,4), (5,3), (3,5), (5, 2)) gives, by formula (16), the five-part sum system which generates the integers 0, 1, 2, . .…”

“…[2] Theorem 9). As shown in Lemma 3.1 and Theorem 4.1 of [5] (see also [2] Theorem 1), a 2ν × 2ν matrix M will have all rows and columns adding up to the same number, and also the associated symmetry described above, if, after subtracting the weight w, it has the form…”

“…1,2,3,4,5,6, 30240, 30241, 30242, 30243, 30244, 30245, 30246, 60480, 60481, 60482, 60483, 60484, 60485, 60486, 90720, 90721, 90722, 90723, 90724, 90725, 90726}, A 2 = {0, 7, 14, 21, 224, 231, 238, 245, 448, 455, 462, 469, 672, 679, 686, 693, 896, 903, 910, 917}, A 3 = {0, 56, 10080, 10136, 20160, 20216, 362880, 362936, 372960, 373016, 383040, 383096, 725760, 725816, 735840, 735896, 745920, 745976, 1088640, 1088696, 1098720, 1098776, 1108800, 1108856, 1451520, 1451576, 1461600, 1461656, 1471680, 1471736}, A 4 = {0, 112, 1120, 1232, 2240, 2352, 3360, 3472, 4480, 4592, 5600, 5712, 6720, 6832, 7840, 7952, 8960, 9072}, A 5 = {0, 28, 120960, 120988, 241920, 241948, 1814400, 1814428, 1935360, 1935388, 2056320, 2056348}, which generates the integers 0, 1, 2, . .…”