Let $A$ be a hermitian matrix. Then all its eigenvalues are real. Let $\lambda_1 \geq \lambda_2 \geq ... \geq \lambda_n$ be the eigenvalues with associated eigenvectors $v_1, v_2. ..., v_n$, respectively.
I'd like to prove that $\displaystyle \max_{0\neq x\perp v_1} \Bigl({x^*Ax\over x^*x}\Bigr)$ exists and that $\displaystyle \max_{0\neq x\perp v_1} \Bigl({x^*Ax\over x^*x}\Bigr)=\lambda_2$.
Generalize the previous statement.
There's a suggestion to take $x=\alpha_1 v_1 + ...+ \alpha_n v_n$. Still can't make anything out of it.
Thanks.