Program, Semester Week 11#
The Eigenvalue Problem and Diagonalization#
A special type of linear map is one that maps from a given vector space \(V\) into itself. For such maps it is possible that certain inputs map to themselves or to scaled versions (scalar multiples) of themselves. These quite special input vectors are called eigenvectors and they come with properties like eigenvalues, which is the resulting scaling factor, corresponding multiplicities and are organized in eigenspaces. The task of determining all these properties is called the eigenvalue problem.
If sufficiently many linearly independent eigenvectors exist within \(V\), then it is possible to choose an ordered basis for \(V\) consisting of purely eigenvectors. In such an eigenbasis, the mapping matrix of the map turns out to be particularly simple to work with - it turns out to be a diagonal matrix. Of that reason we will this week study this related topic of diagonalization as well.
Key Terms#
Eigenvalue \(\lambda\). Eigenvector. Eigenspace \(E_\lambda\). Algebraic multiplicity \(\mathrm{am}(\lambda)\) and geometric multiplicity \(\mathrm{gm}(\lambda)\). Diagonalization.
SymPy Demos#
For this week we have the following two SymPy demos:
Preparation and Syllabus#
This week will cover Sections 11.1 through 11.3 from Chapter [11 - The eigenvalue problem and diagonalization]. Note that Sections 11.4 and 11.5 in this chapter are not a part of our syllabus.
Exercises#
Exercises for Long Day.
Exercises for Short Day.