Kernel PCA and K largest eigenvectors
Let us say $K$ is an $m \times n$ matrix.
You want to find a vector $a$ that maximizes $\frac{||Ka||}{||a||}$.
Now maximizing $\frac{||Ka||}{||a||}$ implies maximizing its square:
$$\begin{eqnarray*}
\frac{||Ka||^2}{||a||^2} \\
= \frac{(Ka)^TKa}{a^Ta} \\
= \frac{a^TK^T Ka}{a^Ta} \\
= \frac{a^T(K^TK) a}{a^Ta} \\
= \frac{a^TSa}{a^Ta}
\end{eqnarray*}
$$
where $S = K^TK$ is always an $n \times n$ symmetric positive semidefinite matrix for all $K$
Now symmetric positive semidefinite matrices have several special properties one of which is that they always have REAL eigen values. Also the eigen vectors associated with these eigen values are always orthogonal to each other.
What this means is that if you locate the eigen vector for $S$ with the highest eigen value - you have basically found the vector that maximizes:
$\frac{a^TSa}{a^Ta}$
So in essence the eigen vector with the largest eigen value for $S$ or $K^TK$ maximizes both $\frac{a^TSa}{a^Ta}$ and the original $\frac{||Ka||}{||a||}$
Use the same analogy for the remainder eigen vectors - i.e the second best vector that maximizes $\frac{a^TSa}{a^Ta}$ is the eigen vector corresponding to the second highest eigen value.