Further results on Hermitian algebras derived from latin squares with potential applications to physics and chemistry: Unitary matrices that diagonalize the algebras

In an earlier paper Srivastava [2005. Combinatorial Hermitian Algebras Derived from Latin Squares. J. Statist. Plann. Inference 129, 305-316], subsequently referred to as [S05], the author had presented an infinite class of algebras of Hermitian matrices. Each of these algebras were derived from Latin Squares. In this paper, we show how to obtain a unitary matrix U (over the complex field) such that (U˜MU) is a diagonal matrix, for all matrices M belonging to a particular algebra. The notation (˜) is defined in the next paragraph. The said diagonal matrix is a matrix that contains the characteristic roots of M. The matrix U may be different for different algebras. Many other related results are also presented. The algebras are further discussed. How the Hermitian matrices arise in Quantum Mechanics (where every observable attribute of every physical phenomenon is in a sense characterized by a Hermitian matrix) is briefly described. The significance of the existence of Hermitian algebras and the application to Physics and Chemistry is pointed out.