A maximization problem and its application to canonical correlation

Let [Sigma] be an n - n positive definite matrix with eigenvalues [lambda]1 >= [lambda]2 >= ... >= [lambda]n > 0 and let M = {x, y x [epsilon] Rn, y [epsilon] Rn, x [not equal to] 0, y [not equal to] 0, x'y = 0}. Then for x, y in M, we have that x'[Sigma]y/(x'[Sigma]xy'[Sigma]y)1/2 <= ([lambda]1 - [lambda]n)/([lambda]1 + [lambda]n) and the inequality is sharp. If is a partitioning of [Sigma], let [theta]1 be the largest canonical correlation coefficient. The above result yields [theta]1 <= ([lambda]1 - [lambda]n)/([lambda]1 + [lambda]n).