Maximize $ | x^\top A x | $ and $x$ is a unit vector

Question

$ A $ is a square matrix and $x$ is a unit vector. We want to maximize $ | x^\top A x | $ for $x$. The solutions I saw so far are based on calculus and Lagrange Multipliers. Is it possible to solve this problem with linear algebra arguments ? I can definitely do that if $A$ is symmetric, in this case I can represent $x$ as a linear combination of an orgthonormal eigenvectors of $A$ and the rest is easy. I wonder how to solve this problem with linear algebra for any square matrix $A$.