Solving Coupled Eigenvalue equations 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 $A$, $B$, $D$
If you write $I^m$ for an $m \times m$ identity matrix, you can make a big matrix equation out of it $$\left(\begin {array}{c  c}A-\lambda I^m &amp; B\\B^T &amp; D-\mu I^n \end {array}\right)\left(\begin {array}{c}x \\ y \end {array}\right)=0$$    which is a classic eigenvalue problem.  Chapter 11 of Numerical Recipes (free for the obsolete versions) or any numerical analysis text can help you.
Perhaps adding the two equations in order to reeduce it to one equation with eqienvalue (lambda + mu) and Eqienvector x+y is the trick. Then find the inverse using a calculator or standard methods. But there is probably a trick involving the fact that the matrices satisfies the properties they do.