Agenda, Semester Week 11

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 solving 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 mapping matrix of 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#

Long Day will cover:

  • Eigenvalues \(\lambda\) and eigenvectors.

  • Eigenspaces \(E_\lambda\).

  • Algebraic multiplicity \(\mathrm{am}(\lambda)\) and geometric multiplicity \(\mathrm{gm}(\lambda)\).

Short Day will cover:

  • Eigenbasis.

  • Diagonalization.

  • Similar matrices.

Preparation and Syllabus#

Syllabus this week is Chapter 12 - note that its final two sections are optional.

  • For Long Day, read Chapter 12 through Section 12.2.

  • For Short Day, read Section 12.3.

Exercises#

Exercises for Long Day.

Exercises for Short Day.

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