Expectations of Ratios of Quadratic Forms in Normal Variables: Evaluating Some Top-Order Invariant Polynomials

Chikuse's (1987) algorithm constructs top‐order invariant polynomials with multiple matrix arguments. Underlying it is a set of simultaneous equations for which all integer solutions must be found. Each solution represents a component of the sum of terms which comprise the polynomial. The system of equations has a specialised structure which may be exploited to obtain a polynomial with r matrix arguments in terms of a polynomial with r‐1 matrix arguments. This is demonstrated for two particular polynomials that have two matrix arguments. These results are applied to problems involving expectations of ratios of quadratic forme in normal variables; analytic as well as computable formulae are derived.