The Spectral Theorem -I

Abstract

Let A be a Hermitian operator on the space ℂⁿ. Then there exists an orthonormal basis {e _j } of ℂⁿ each of whose elements is an eigenvector of A. We thus have the representation

$$A = \sum\limits_{j = 1}^n {{\lambda _j}\left\langle { \cdot ,{e_j}} \right\rangle {e_j},}$$

((24.1))

where Ae _j = λ _j e _j . We can express this in other ways. Let λ₁ > λ₂ > ⋯ > λ _k be the distinct eigenvalues of A and let m₁, m₂,&, m _k be their multiplicities. Then there exists a unitary operator U such that

$$U*AU = \sum\limits_{j = 1}^k {{\lambda _j}{P_j},}$$

((24.2))

, where P₁, P₂,&, P _k are mutually orthogonal projections and

$$\sum\limits_{j = 1}^k {{P_j} = I.}$$

((24.3))

. The range of P _j , is the m _j -dimensional eigenspace of A corresponding to the eigenvalue λ _j . This is called the spectral theorem for finite-dimensional operators.