BIORTHOGONAL AND DUAL CONFIGURATIONS AND THE RECIPROCAL NORMAL DISTRIBUTION
0. Summary. In this note we discuss the notions of biorthogonal and dual configurations and their relevance in certain statistical applications. The first application is to the distribution of a random matrix related to a multi-variatenormal sample matrix. As with the latter, the distribution is preserved by (certain) linear transformations. One consequence of this is the familiar result that if Q is a non-singular Wishart matrix, then for any non-zero vector a, 1/a'Q-'a is a multiple of a chi-square variable. Application is also made to the Gauss-Markov theorem and to certain estimates of mixing proportions due to Robbins. 1. Biorthogonal and dual configurations. Let S be a vector space with an inner-product, denoted by (. , .). The configurations (= ordered subsets) (xi, * - *, xp) and (xi*, ... *, x*) are said to be biorthogonal if (xi, xj*) = aij, the Kroneker delta. Clearly this relation is symmetric. Necessarily, the elements of {xi, * **, xp.} (respectively, {xi*, *. *, xp}) are linearly independent. For if (e.g.) xi E 'U {x2, * * *, xp,}, the subspace spanned by {x2, * * *, x}, then (xi, xi*) = 0, i = 2, ... , p, implies that (xi, xi*) = 0. In general, there are many configurations biorthogonal with a given configuration (xi, **.*, x One such is distinguished: There is a unique (y,, * **, y,) C 'U {xl, *. , xP} which is biorthogonal with (xi, * * *, xp). It is called the configuration dual to (xi, * - *, xp) and is constructed as follows: Let xi, be the projection of xi into 'L{xi, * , Xj_1, xi+1, ... , xp}. (q'L is the orthogonal complement of cl C SC in S.) (xi, xi.) = (xi., xi.) 0 0 by linear independence; of course (xi, xj.) = 0 if i 0 j. Then yi = xi.l(xi., xi.) gives the configuration dual to (xi, * * *, xp). It readily follows that any configuration (xi*, ..., xp*) biorthogonal with (xi, * , xp,) has the representation xi* = yi + ai, where t5 ? 'UL{Xi, ... Xp}. Since yi'U E x, **, xp}, we may write (y,, ***, yp) = (xi, xp)A,I where A is a non-singular p X p matrix. Letting Q denote the non-singular configuration matrix of (xi, *.* , xp) :qij = (xi, xj), the duality relations require that QA = I. Hence A = Q-1, (y,, * * , y,,) = (xi, * ,xp) Q' and the dual configuration has configuration matrix Q-1. Thus the dual configuration relation is also symmetric. If xl, * * *, xp are elements of RP and if X = ( (xl, * * *, xp) is the p X p matrix they generate by their representation as p-tuples, then Y' =