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11: The Spectral Theorem for normal linear maps

Last updated

Mar 5, 2021
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- 10.E: Exercises for Chapter 10
- 11.1: Self-adjoint or hermitian operators

( \newcommand{\kernel}{\mathrm{null}\,}\)

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 $F = C$ .

11.1: Self-adjoint or hermitian operators
11.2: Normal operators
11.3: Normal operators and the spectral decomposition
11.4: Diagonalization
11.5: Positive operators
11.6: Polar decomposition
11.7: Singular-value decomposition
The singular-value decomposition generalizes the notion of diagonalization.
11.E: Exercises for Chapter 11

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