I want to optimise
$Q(A, B, y, z) = (\mathbf{x}-\mathbf{Ay}-\mathbf{Bz})^T\mathbf{\Sigma}^{-1}(\mathbf{x}-\mathbf{Ay}-\mathbf{Bz})$
with the constraint
$\mathbf{A}^T\mathbf{\Sigma}^{-1}\mathbf{B}=\mathbf{0}$.
Essentially I want to find optimums for A, B, y and z, so I thought of using a Lagrange multiplier, e.g.:
$Q(A,B,y,z) - \lambda(\mathbf{A}^T\mathbf{\Sigma}^{-1}\mathbf{B})$
and then differentiate for each of the variables. But the function returns a scalar and the constraint is a vector, so this doesn't make sense.. I guess I can consider a vector of Lagrange multipliers, but how would they relate to the matrix in the constraint? Inner product? I can't find a good source for this (or not sure what exactly to search for!).