I wish to solve the following set of coupled eigenvalue equations. How should I do it?
For real matrices $A$,$B$,$D$ and vectors $x \in R^m$, $y \in R^n$ $ A x + B y = \lambda x $ $ B^T x + D y = \mu y $ where, $A$ and $D$ are symmetric.
Background:
I am trying to solve the following optimization problem:
$ \min x^T A x + y^T D y + x^T By $ $ \textrm{such that } \, x^T x = 1 , y^T y = 1 $ This leads to the above eigenvalue problem. $\lambda$ and $\mu$ are the Langrangian parameters for the constraints.
EDIT: I need to actually find numerical solutions for these equations given $A$, $B$, $D$