Lattice

0.1 Hermitian and Unitary Matrices1JJJ

Definition 0.1.1  A matrix $A\in\mathbb{C}^{n\times n}$ is Hermitian iff $A^*=A$.

Definition 0.1.2  A matrix $A\in\mathbb{C}^{n\times n}$ is unitary iff $A^*A=AA^*=I$.

Proposition 0.1.3  If $Q\in\mathbb{C}^{n\times n}$ is unitary, then

1. $||Q\mathbf{u}||=||\mathbf{u}||, \quad\forall \mathbf{u}\in\mathbb{C}^n$.
2. The rows and columns of $Q$ are an orthonormal basis of $\mathbb{C}^n$.

Theorem 0.1.4  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 $A$.