On Symmetric, Orthogonal, and Skew-Symmetric Matrices

Introduction and Notation. In this paper all the scalars are real and all matrices are, if not stated to be otherwise,p-rowed square matrices. The diagonal and superdiagonal elements of a symmetric matrix, and the superdiagonal elements of a skew-symmetric matrix, will be called the distinct elements of the respective matrices. Σ will denote both the set of all symmetric matrices and the ½p(p+ 1)-dimensional space whose coordinates are the distinct elements arranged in some specific order.Kwill denote both the set of all skew-symmetric matrices and the ½p(p– 1)-dimensional space whose coordinates are the distinct elements arranged in some specific order. Any sub-set of Σ(K) will mean both the sub-set of symmetric (skew-symmetric) matrices and the set of points of Σ(K). Any point function defined in Σ(K) will be written as a function of a symmetric (skew-symmetric) matrix.Dαwill denote the diagonal matrix whose diagonal elements are α1, α2, …, αp. The characteristic roots of a symmetric matrix will be called its roots.