Agenda, 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 of themselves. Such rather special input vectors are called eigenvectors and with these follow properties like eigenvalues, algebraic multiplicities, geometric multiplicities, and eigenspaces. The task of determining these properties is known as 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. This is known as an eigenbasis, and with respect to eigenbases, a linear map between vector spaces is particularly simple - it becomes a diagonal matrix. We will study this related topic of diagonalization this week as well.

Key Terms

Eigenvalues \(\lambda\) and eigenvectors. Eigenspaces \(E_\lambda\). Algebraic multiplicity \(\mathrm{am}(\lambda)\) and geometric multiplicity \(\mathrm{gm}(\lambda)\). Diagonalization and similar matrices.

Preparation and Syllabus

This week will cover Sections 12.1 through 12.3 from Chapter 12 in [the textbook]. Note that Sections 12.4 and 12.5 in this chapter are not a part of our syllabus.

Exercises

Exercises for Long Day. A SymPy test in Möbius for Python module 3 will take place 15:30-17:00 on Long Day.

Exercises for Short Day.