Before going deep into spectral theory, a quick look into the idea where all these got developed would be necessary, the Hermitian matrices and the normal matrices. It is worth noting that a lot of contents are already covered in the [[Tensor calculus and linear algebra#Self-adjoint transformations and symmetry]]. This note provides a more metric viewpoint of self-adjointness and transformation. For matrix of finite dimension, we call $A$ a Hermitian matrix if $A^{*}= A$ where $A^{*} =\overline{A^{T}}$. We consider the eigenvalue problem of such a matrix, $Au=A\lambda$ where $u\neq0$, $\lambda \in \mathbb{C}$. Some core properties of a Hermitian matrix: 1. All eigenvalues of a Hermitian matrix are real, i.e. $\lambda \in \mathbb{R}.$ 2. A Hermitian matrix admits an orthonormal basis of eigenvectors. 3. A Hermitian matrix can be unitarily diagonalized. That is, there exists a unitary matrix $U$ such that $A = U \Lambda U^{*},$ where $\Lambda$ is diagonal.