I am trying to show that the symmetric normalized Graph Laplacian and symmetric normalized Adjacency matrix have corresponding eigenvalues $\lambda_i$ and $1 - \lambda_i$ for i=1 to n.
$\lambda$ is an eigenvalue of $D^{-1/2} (D- A) D^{-1/2}$ if and only if $1-\lambda$ is an eigenvalue of $D^{-1/2} A D^{-1/2}$. Does someone have any idea where should I start?
This is a quick exercise in matrix algebra:
$$ D^{-1/2} (D- A) D^{-1/2} = I - D^{-1/2} A D^{-1/2} $$
Then notice $\lambda$ is an eigenvalue of $M$ if and only if $1-\lambda$ is an eigenvalue of $I-M$. In fact the eigenvectors corresponding to $\lambda$ for $M$ are the same as the eigenvectors corresponding to $1-\lambda$ for $I-M$, and vice versa.