Decomposition of Symmetric Matrices and Distributions of Quadratic Forms

Abstract

This paper presents some theorems for real matrices which should be of use and interest to anyone working with quadratic forms in normally distributed variables because of the following equivalences: (i) stochastic independence of forms and orthogonality of their matrices (Craig [3] and Carpenter [1]), (ii) $\chi^2$ distribution of a form and idempotency of its matrix (Carpenter [1]), and (iii) the distribution of a form as the unique difference of two stochastically independent $\chi^2$ distributions and the tripotency of its matrix (i.e., $A = A^3$). The last equivalence is a consequence of the first two and the fact that every symmetric matrix is the unique difference of two non-negative, orthogonal symmetric matrices. The above equivalences apply specifically to forms in independently distributed variables with common variance, but this is inconsequential since if the random vector $X$ has covariance matrix $V$, then $Y = V^{-\frac{1}{2}}X$ has covariance matrix $I$ and $X'AX = Y'V^\frac{1}{2} AV^\frac{1}{2}Y$. Hence the matrix theorems of this paper can be easily adjusted to fit the more general case. The matrix theorems of this paper emphasize the use of the trace of a matrix, a concept which has not been fully exploited in such results as these. As an introductory remark, we mention that throughout this manuscript all matrices are assumed to be symmetric except for those involved in Lemma 3; and our application of that lemma will be to symmetric matrices only.