Extrema of sums of heterogeneous quadratic forms

We analyze the following problem arising in various situations in multivariate statistical analysis. We are given k symmetric, positive definite n × n matrices, A1, A2,…, Ak (k n), and we would like to maximize the function ∑ki = 1 xTiAixi under the constraint that x1, x2,…, xk Rn form an orthonormal system. Some theoretical results as well as an algorithm are presented.