Simultaneous reduction of a pair of quadratic forms

Abstract

Given two quadratic forms Q₁=x'Ax and Q₂=x'Bx one of which (say Q₂) is p.d., it is well known that both are simultaneously reducible to forms containing square terms only by a real non-singular transformation and also by contragradient transformations. In this paper necessary and sufficient conditions are obtained for other cases such as (a) Q₁ arbitrary, Q₂ n.n.d., (b) Q₁ arbitrary, Q₂ non singular, and (c) Q₁ and Q₂ are both arbitrary. For a pair of symmetric matrices A and B of the same order and B n.n.d., the paper introduces the concept of a proper eigen value and a proper eigen vector of A with respect to B. Such eigen vectors are shown to determine the required transformation for the simultaneous reduction of associated quadratic forms.