Theorem 1.2 (Spectral theorem for Hermitian matrices). Assuming that:

$Ω$ finite
$M : Ω \times Ω \to ℂ$ Hermitian
$| Ω | = n$

Then there exist

λ_{1}, \dots, λ_{n} \in ℝ

and

φ_{1}, \dots, φ) n \in l^{2} (Ω)

non-zero such that

(1)
$M φ_{i} = λ_{i} φ_{i}$
(2)
$⟨ φ_{i}, φ_{j} ⟩ 𝟙_{i = j}$
(3)
$M = \sum_{i = 1}^{n} λ φ_{i} φ_{i}^{H}$
(4)
there exists $U$ orthogonal such that $U M U^{H} = diag (λ_{i})$
(5)
if $M$ is real, then can take $φ$ to be real ( $φ : Ω \to ℝ$ )