On this page, we discuss some fundamental ideas about eigenvalues and eigenvectors of a matrix, which is sometimes called Spectral Theory.
Note: The basic and advanced learning objectives listed below are meant to give you an idea of the material you should learn about this section. These are mainly intended to be used in a course which uses an Active Learning approach, where students are required to "read ahead" before each class - but can equally be used in a more traditional course setting.
Unless your teacher gives you specific instructions, it is up to you to decide how much of the listed resources you need to read or watch - you probably do not need to go through all of it. You might also want to look at the General Study Tips & Tricks page for some recommendations on how to effectively study with a math textbook and videos.
Basic learning objectives
These are the tasks you should be able to perform with reasonable fluency when you arrive at your next class meeting. Important new vocabulary words are indicated in italics.
- Verify if a given vector is an eigenvector of a matrix, and find its associated eigenvalue if applicable.
- Find the characteristic polynomial of a matrix.
- Find the eigenvalues of a matrix, by finding the roots of its characteristic polynomial.
Advanced learning objectives
In addition to mastering the basic objectives, here are the tasks you should be able to perform after class, with practice:
- Find the eigenspace (and eigenvectors) for each eigenvalue of a matrix.
To prepare for class
Watch the first 13 minutes of this video by 3Blue1Brown which explains the motivation behind eigenvalues and eigenvectors and gives a good introduction on how to find them (if they exist) - ignore the last topic ("Eigenbasis") for now:
Watch these videos by Khan Academy which goes through a detailed example of finding eigenvalues and eigenvectors for a \(3\times 3\) matrix: