Spectral Theorem for Self-adjoint Linear Operators

Let V be a finite-dimensional vector space, either real or complex, and equipped with an inner product 〈· , ·〉. Let A : V → V be a linear operator. Recall that the adjoint of A is the linear operator A : V → V characterized by 〈Av, w〉 = 〈v, Aw〉 ∀v, w ∈ V (0.1) A is called self-adjoint (or Hermitian) when A = A. Spectral Theorem. If A is self-adjoint then there is an orthonormal basis (o.n.b.) of V consisting of eigenvectors of A. Each eigenvalue is real. Before proceeding to the proof, let us note why this theorem is important. Recall that, after making a choice of of basis v1, . . . , vn for V , a linear operator A corresponds to an n× n matrix A = (Aij) where n = dimV . The correspondence is given by Avj = n