Let $A$ be an adjacency matrix of undirected graph $G$, where $G$ is a connected graph. The normalized adjacency matrix is defined as $\hat{A}=D^{-1/2}AD^{-1/2}$, where $D$ is degree matrix of graph $G$. Note that the adjacency matrix $A$ does not contain a self-loop. Is there a matrix $P$ composed of the eigenvectors of matrix $\hat{A}$ and a diagonal matrix $C$ composed of the eigenvalues of matrix $\hat{A}$, such that $\hat{A}=PCP^{-1}$?
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1$\begingroup$ Are you just asking whether $\hat A$ is diagonalisable? If so, the answer is that symmetric matrices are always diagonalisable. $\endgroup$Dave Benson– Dave Benson2023-04-08 14:20:33 +00:00Commented Apr 8, 2023 at 14:20
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