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# 11: The Spectral Theorem for normal linear maps

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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

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

This page titled 11: The Spectral Theorem for normal linear maps is shared under a not declared license and was authored, remixed, and/or curated by Isaiah Lankham, Bruno Nachtergaele, & Anne Schilling.