On the existence of a special type of symmetric matrix and the construction of Hadamard matrices

Abstract

In this paper  we consider a symmetric matrix $A^{2} $ which is the square of an unknown matrix $A$ with only the two numbers $+1$ and $-1$ as its entries and we establish the existence of a special type of square matrix $A^{2} $. From this special square matrix $A^{2} $ all the possible matrices $A$ can be obtained and used for the construction of Hadamard matrices. These Hadamard matrices are much useful in coding theory, communication theory, signal processing and cryptography.