**Theorem 0.0.1 **  **Eigenvalue Decomposition of Hermitian Matrices Theorem** states that any Hermitian matrix $A\in\mathbb{C}^{n\times n}$ can be diagonalized by an Hermitian matrix $Q\in\mathbb{C}^{n\times n}$ that is
\[A = Q\begin{pmatrix}\lambda_1 && \ &&\ \\ \ && \ddots&&\ \\\ &&\ &&\lambda_n\end{pmatrix}Q^*\]
where $\lambda_1,\dots,\lambda_n$ are the eigenvalues of $A$ and the columns of $Q$ are eigenvectors of $A$.