Let's consider S a square, symmetric matrix, why the maximum eigenvalue is given by: $$ \lambda_{max}(S) = \max_{x: ||x||_2=1} x^T S x$$
By the EVD of symmetric matrices, I will agree if we restricted $x$ to be an eigenvector because:
$$ u_i^T S u_i = \lambda_i ||u_i||^2 $$
but I seem to miss a step to get the result without the restriction for $x$ to be an eigenvector.