I would like to shed some light on the following problem:
$$A_2 x = \lambda \odot A_1x$$
where $A_1$ and $A_2$ are known square matrices and $\odot$ is the element-wise operator. This generalize the notion of eigenvalue decomposition, where now the scaling $\lambda$ is a vector instead of a scalar value. Is there any sort of literature on this problem and relative solutions? Thanks.
$(\lambda)_i = (A_2 x)_i / (A_1 x)_i$ is always a solution. The only possible complication is if the $i$th component of $A_1 x$ is zero.