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

Extend the selected ${\ell }_1$ to an orthonormal basis of eigenvectors, writing it as $v_1$. For a unit vector orthogonal to it, $a_1=0$, and thereforeA unit eigenvector $v_2$ attains equality. Thus ${\ell }_2$ is a unit eigenvector for the largest eigenvalue on ${\ell }_1^{\perp }$, and the maximum is ${\lambda }_2$. If the largest eigenvalue is repeated, a second orthogonal vector in that eigenspace is allowed and ${\lambda }_2={\lambda }_1$; there is no uniquely determined first pair of directions.