I am solving a problem of the form
$$ A\frac{d^2\Phi}{dy^2} + B\frac{d\Phi}{dy} + C\Phi = 0 $$
where phi is the eigenvector solution and A, B, and C are n x n square matrices. If I know matrices A, B, and C and know a particular eigenvalue and corresonding eigenvector for this equation is there a way to determine the eigenvector solution of the adjoint homogeneous problem using the value/vector that I already know?