Bi-orthogonality theorem

The bi-orthogonality theorem connected with the eigenvalue problems for a pair of operators T and Td. 

This is the general bi-orthogonality theorem valid for a pair of adjoint operators. From the expansion theorem x = Xk Ck ak, one gets further [Di x] = Ek [Di CJak = [Di Ci]ai. Since the product [Di x] cannot be... 

We note that, due to the bi-orthogonality theorem, the metric matrix A = < C I C > is automatically block-diagonalized with different blocks for different eigenvalues, and it is then possible to treat the stable subspaces associated with different eigenvalues seperately, which is an essential simplification. In addition, the symmetry property (2.24) indicates another simplification from the computational point of view. 

---