Given a matrix $A \in \Bbb R^{d \times d}$, why is the following identity true?
$$ \lambda_0 := \max_{x \in \Bbb R^d, \|x\|_2 = 1} x^T A x $$
where $\lambda_0$ is the largest eigenvalue of $A$.
Also, does the solution to the above equation has $x$ in the same direction as the eigenvector corresponding to $\lambda_0$? What connections does the identity have with any of the norms of $A$?
I have tried writing $A$ with its eigen decomposed form, i.e. $A=QDQ^{-1}$ hoping that I would be able to get a form of $y^TDy$ which would mean the RHS can be written as a linear combination of the eigenvalues of $A$. But I am stuck after the first step.
This statement is true if $A$ is symmetric as a consequence of the spectral theorem, but it is not true more generally. As an example, consider the diagonalizable matrix $$ A = \pmatrix{34&24\\0&41}. $$ We have $\lambda_0 = 41$, but the maximum of $x^TAx = 50$ is attained with $x = \frac 15(3,4)^T$.