Hypermatrix and its Application
Kin-ichi Yamada · Institutional Repositories DataBase (IRDB) · 1965
In this paper, the author will cons der multld mensronal matrlces on a Ime dlfferent from that of R. Gouarn and I. Samuel.1 In the first half, a singular situation shall be pointed out concerning multidimensional matrices and in the second half, an application of a specific kind of them shall be proposed. S O HypermatricesA setout of numbers on the lattice points (i,j, ...,k,1) of an N-dimensional space will be called an .N:-dimensional matrix of order IXJX ... XKXL or an IXJX ... XKXL hypermatrix, where i ranges over 1, 2, ..., I, j over 1, 2, ..., J, ..., k over 1, 2, .,., K and I over 1, 2, ..., L.The number a*j.hl set out on the point (i,j, ..., k, l) will be called the i･j･... ･k･1 element of the hypermatrix, which as a whole will be denoted by A or by [aij ht].The elements shall be real numbers and the dimension be fixed hereafter.Equality and inequality of two hypermatrices of the same order will be defined in the same way as usual.Thus they enjoy the usual fundamental properties : reflexivity, symmetricity or antisymmetricity and transitivity.Addition of two hypermatrices of the same order will be defined also as usual, and it enjoys commutativity and associativity.There exists the additive identity and every hypermatrix has its additive reciprocal.Scalar multiplication of a hypermatrix by a real number will be defined too as usual.And it enjoys commutativity, associativity and double distributivity.Let A= [aiJ,kl] be an IXJX...XKXL hypermatrix and B=[bpq .*] a PXQX...XRXS hypermatrix with L=P.Then multiplication of A by B shall be defined by AB=[ aij .klblq."s]' l,q,.. ,' Thus it proves easily to be associative and doubly distributive over addition.S I Cubic Hypermatnces An LXLX ... X LXL hypermatrix will be called cubic.The set of all cubic hypermatrices with a specific L forms a ring with respect to addition and multiplication defined above, since it is closed under these operations which possess the fundamental properties as was shown * Professor (A-"o ju) in Mathematics.l Ren6 Gouarn et Isaac Samuel : Introduction l' tude des matrices multldimensionnelles. Cahiers