I'm reading a lecture on clustering (https://www.math.ucdavis.edu/~strohmer/courses/180BigData/180lecture_clustering.pdf)
And I was wondering about the proofs for Theorem 3.3 and 3.4.
That is,
For two normalized versions of the graph Laplacian, a symmetric one and a non-symmetric one,
$L_s = D^{-\frac{1}{2}}LD^{-\frac{1}{2}}=I-D^{-\frac{1}{2}}WD^{-\frac{1}{2}}$
$L_N=D^{-1}L=I-D^{-1}W$
We have that
(3.3) $\lambda$ is an eigenvalue of $L_N$ with eigenvector $u$ if and only if $\lambda$ and $u$ solve the generalized eigenproblem $Lu=\lambda Du$
and
(3.4) $0$ is an eigenvalue of $L_N$ and the associated eigenvector is $1$. $0$ is an eigenvalue of $L_S$ and the associated eigenvector is $D^{\frac{1}{2}}1$.
How would I prove these two statements?