Matrix Functions

Let \(H\) be an arbitrary Hermitian matrix on \(\mathbb{C}^{d}\) and \(f: \mathbb{R} \mapsto \mathbb{R}\) be a real valued function. Since \(H\) can be unitarily diagonalized,

\[ H = U D U^{\dagger} = U \begin{pmatrix}\lambda_{1}\\&\lambda_{2}\\&&\ddots\\&&&\lambda_{d}\end{pmatrix} U^{\dagger} \]

Therefore, applying function \(f\) has the effect

\[ f(H) = U \begin{pmatrix}f(\lambda_{1})\\&f(\lambda_{2})\\&&\ddots\\&&&f(\lambda_{d})\end{pmatrix} U^{\dagger} \]