Perator Decompositions

1 OPERATOR DECOMPOSITIONS In this lecture we will consider two basic decompositions of operators that will be used repeatedly in the course: the spectral decomposition and the singular-value decomposition. The spectral decomposition holds only for normal operators, while the singular-value decomposition holds for all operators. The singular-value decomposition will be particularly important in the section following this one, where it will be closely related to various norms on operators. Let us begin with the Spectral Theorem, which holds for any normal operator A ∈ L (X) acting on a complex Euclidean space X. Recall that A is said to be normal if AA * = A * A. Theorem 1 (Spectral Theorem). Let X be an n-dimensional complex Euclidean space, let A ∈ L (X) be a normal operator, and assume that spec(A) = {λ