Multiplicative Matrix & Generalized Idempotent Matrix & Applications

Hadamard conjectured that such a matrix exists of every 4th order when it is a positive integer. The conjectured remains unsettled even today. Several mathematician forwarded methods for constructing such matrices known as Hadamard matrices. Later on, the notion of Hadamard matrices is generalized by taking group elements in place of +1 and -1 as its entries. The present book is motivated by the Hadamard work and other mathematicians who forwarded his work. We have constructed with entries +1, -1 satisfying M2 = m M, 1¿m¿n. We call this matrix as a generalized idempotent matrix. The purpose of this book is to construct, enumerate and study the certain type of combinatoric matrices and their applications. The generalization of the idempotent matrix is a quite new concept. It has several uses in the theory of error correcting codes, encryption etc. In Chapter I some basic terms and definitions are given and they are used in later Chapters. In Chapter II we have constructed nxn generalized idempotent matrices M with entries 1, -1 satisfying M2 = mM, 1 m n.