C.2 The eigenvalue equation in kernel PCA

The equivalence of the equations

and

is shown.

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 ,
= *u*_{l}^{l}. At least, two *u*_{l} must differ from zero because
itself is not an eigenvector. It follows that
= *u*_{l}^{l} 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).

2005-03-22