The equivalence of the equations
Equation (C.8) follows immediately from (C.9). To prove the opposite direction, we assume that a vector exists that is not an eigenvector of , while is an eigenvector of . This assumption infers that (C.8) is fulfilled and (C.9) is not. Thus, we need to show that the assumption leads to a contradiction.
We only consider the case that is orthogonal to the subspace ker (the space of vectors fulfilling = 0) because the elements of ker --if they exist--solve already both (C.8) and (C.9). Since is symmetric, can be written as a linear combination of the pairwise orthogonal eigenvectors l of , = ull. At least, two ul must differ from zero because itself is not an eigenvector. It follows that = ull with being the eigenvalues corresponding to l. All eigenvalues are non-zero because is orthogonal to ker . Thus, can be also not an eigenvector of . This contradicts our first assumption. Therefore, (C.9) follows from (C.8).