Math 108b: Notes on the Spectral Theorem

For a general vector space V , and a linear operator T , we have already asked the question “when is there a basis of V consisting only of eigenvectors of T?” – this is exactly when T is diagonalizable. Now, for an inner product space V , we know how to check whether vectors are orthogonal, and we know how to define the norms of vectors, so we can ask “when is there an orthonormal basis of V consisting only of eigenvectors of T?” Clearly, if there is such a basis, T is diagonalizable – and moreover, eigenvectors with distinct eigenvalues must be orthogonal.