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Mathematics LibreTexts

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 \(\mathbb{F} = \mathbb{C}\).

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