Past exam of the mathematics course of the University of Cambridge / 2005 / iii / Paper 46 / 2 / i / Solution

Let ${\lambda }_1\ge \cdots \ge {\lambda }_p\ge 0$ be the eigenvalues of the symmetric covariance matrix $V$, and choose an orthonormal basis of eigenvectors $v_1,\dots ,v_p$. If $\ell ={\sum }_j{a}_j{v}_j$ has unit length, then ${\sum }_j{a}_j^2=1$ andEquality is attained by $v_1$, and precisely by unit vectors in the top eigenspace. Hence choose ${\ell }_1$ to be any unit eigenvector for the largest eigenvalue; the maximum variance is ${\lambda }_1$. This proves the maximizing property of the Rayleigh quotient, rather than merely specifying a stationary point.