Stationary values of the ratio of quadratic forms subject to linear constraints

Abstract

Let A be a real symmetric matrix of order n, B a real symmetric positive definite matrix of order n, and C an n$\times$p matrix of rank r with r $\leq$ p > n. We wish to determine vectors $\underset ~\to x$ for which ${\underset ~\to x}^T\ A\underset ~\to x\ / {\underset ~\to x}^T\ B\underset ~\to x$ is stationary and $C^T \underset ~\to x\ = \underset ~\to \Theta$, the null vector. An algorithm is given for generating a symmetric eigensystem whose eigenvalues are the stationary values and for determining the vectors $\underset ~\to x$. Several Algol procedures are included.