ON SELF-INVERSE BINARY MATRICES OVER THE BINARY GALOIS FIELD

Abstract

An important class of square binary matrices over the simplest finite or Galois Field GF(2) is the class of involutory or self-inverse (SI) matrices. These matrices are of significant utility in prominent engineering applications such as the study of the Preparata Transformation or the analysis of synchronous Boolean Networks. Therefore, it is essential to devise appropriate methods, not only for understanding the properties of these matrices, but also for characterizing and constructing them. We survey square binary matrices of orders 1, 2 and 3 to identify primitive SI matrices among them. Larger SI matrices are constructed as (a) the direct sum, or (b) the Kronecker product, of smaller ones. Illustrative examples are given to demonstrate the construction and properties of binary SI matrices. The intersection of the sets of SI and permutation binary matrices is studied. We also study higher-order SI binary matrices and describe them via recursive relations or Kronecker products. Our work culminates in an exposition of the two most common representations of Boolean functions via two types of Boolean SI matrices. A better understanding of the properties and methods of constructing SI binary matrices over GF (2) is achieved. A clearer picture is attained about the utility of binary matrices in the representation of Boolean functions.