For some real vector $x \in \mathbb{R}^p$ and $p \times p$ positive definite real matrices $A_1, A_2, \dots, A _n$, consider another vector of quadratic forms:
$$ (x^\top A_1 x, \dots, x^\top A_n x). $$
I am trying the get the maximum element of the $n$ quadratic forms. Once I have the maximum element, I do not need the rest. Since I have to repeat the computation many times with different $x$, I have been thinking about fast and efficient ways to achieve this goal.
Is there a way to find the maximum without computing all of them? I tried to diagonalize the matrices to use the eigenvalues but don't know how to proceed.