Suppose $A$ is an $n$-by-$n$ symmetric matrix, and I want to find $x_{1}$ and $x_{2}$ that maximize $x_{1}^{T} A x_{1} + x_{2}^{T} A x_{2}$ subject to the constraint that $x_{i}^{T} x_{j} = \delta_{ij}$, where $\delta_{ij}$ is the Kronecker delta function (in other words, $x_{1}$ and $x_{2}$ must be orthogonal unit vectors). I know that the answer to this problem is to let $x_{1}$ and $x_{2}$ be orthonormal eigenvectors of $A$ corresponding to the two largest eigenvalues. I have found this result cited in a couple of places on the internet, but cannot seem to find a proof. I'm also not sure what this result is called, although some people call it an extremal trace problem. However, I cannot find other source that use the phrase "extremal trace problem."
So, what are these problems called? And how can I prove that the solution is given by the eigenvectors corresponding to the largest eigenvalues?