On an Application of Kronecker Product of Matrices to Statistical Designs

Abstract

By a statistical design (or simply, a design) we mean an arrangement of a certain number of "treatments" in a certain number of "blocks" in such a way that some prescribed combinatorial conditions are fulfilled. With every design is associated a unique matrix called the incidence matrix of the design (definitions, etc., in subsequent sections). In many instances, e.g., [7] [8], [10], [12], [16], information regarding certain kinds of designs such as BIB, PBIB designs is obtained from properties of the matrix $NN'$ or of its determinant $|NN'|$ where $N$ is the incidence matrix of the design under consideration. On the other hand in a few cases, such as [4], [5], [11], [14], [15], the incidence matrix $N$ itself has been to investigate properties of designs. This paper gives a method of using incidence matrices of known designs to obtain new designs. In Section 2 we have defined the Kronecker product of matrices. This definition and some properties of the Kronecker product of matrices are given in [1]. Section 3 is devoted to a general discussion of an application of the concept of the Kronecker product of matrices to define the Krnoecker product of designs. This section also contains two theorems which illustrate the use of the method of obtaining Kronecker products of designs. Definitions of some well-known designs are given in Section 4, which also contains a number of results giving explicit forms of certain Kronecker products. Finally some illustrations of a few results of Section 4 are given in Section 5.