On Symmetric, Orthogonal, and Skew-Symmetric Matrices

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Abstract
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. K will 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. Dα will denote the diagonal matrix whose diagonal elements are α1, α2, …, αp. The characteristic roots of a symmetric matrix will be called its roots.

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