The Spectral Theorem
In the last two weeks of the course we will discuss the spectral theorem. Since this is not covered in Dummit and Foote, we will follow Sheldon Axler’s Linear Algebra Done Right (4th edition, draft version) distributed in class. Thanks to Professor Axler for making this available to us.
The relevant sections are:
- Chapter 5, Section A: Eigenvalues and Eigenvectors
- Chapter 6
- Chapter 7, Sections A, B
If time permits we will discuss the singular value decomposition (Chapter 7, Section E).
Eigenvalues and Eigenvectors
Let
If there is a basis of
The spectral theorem (real matrix version)
Theorem: Let
where the are real numbers when , where is the “dot product” of vectors in .
Inner Products
Now we look at the generalization of the dot product.
Let
Then a (real) inner product on
Over
Then
Such a map is sometimes called Hermitian or conjugate linear.
If
is real and nonnegative for all , and is zero only when . is complex-linear as a function of its first variable. .
Definition: An inner product space is a finite dimensional real or complex vector space with a specified inner product.
The spectral theorem (for real inner product spaces)
Definition: Let
Theorem: Suppose that
Note: If the inner product is the usual dot product, then the adjoint to a linear map is its tranpose; therefore “self-adjoint” means “symmetric” in that situation.