# 11. The Spectral Theorem for normal linear maps

In this chapter we come back to the question of when a linear operator on an inner product space \(V\) is diagonalizable. We ﬁrst introduce the notion of the adjoint (a.k.a. Hermitian conjugate) of an operator, and we then use this to deﬁne so-called normal operators. The main result of this chapter is the Spectral Theorem, which states that normal operators are diagonal with respect to an orthonormal basis. We use this to show that normal operators are “unitarily diagonalizable” and generalize this notion to ﬁnding the singular-value decomposition of an operator. In this chapter, we will always assume \(\mathbb{F} = \mathbb{C}\).

### Contributors

- Isaiah Lankham, Mathematics Department at UC Davis
- Bruno Nachtergaele, Mathematics Department at UC Davis
- Anne Schilling, Mathematics Department at UC Davis

Both hardbound and softbound versions of this textbook are available online at WorldScientific.com.