Suppose we are given a set of symmetric, positive definite matrices $A_1,A_2,\ldots,A_k\in\mathbb{R}^{n\times n}$. Is there any numerical method or reduction to a known problem (e.g. eigenvalue computation) that can solve the following problem?
$$ \begin{array}{rl} \min_{v_1,\ldots,v_k\in\mathbb{R}^n} & \sum_{i=1}^kv_i^\top A_iv_i\\ \textrm{subject to }&v_i^\top v_j=\delta_{[i=j]}. \end{array} $$
Note: The problem has some resemblance to the "joint diagonalization" problem, e.g. in this paper. The difference seems to be that in joint diagonalization you use the same basis to diagonalize multiple matrices.
Note 2: If the $A_i$'s do not have the same eigenvectors, then the $v_i$'s will not be exactly eigenvectors of the $A_i$'s.
Note 3: My intuition is that this will give a set of mutually orthonormal vectors $v_1,\ldots,v_k\in\mathbb{R}^n$, with the property that vector $v_i$ is similar to an eigenvector of $A_i$. But this is not an eigenvalue problem.