Dave4Math / Courses / Eigenvalues and Eigenvectors

Eigenvalues and Eigenvectors

Master eigenvalues and eigenvectors in this linear algebra course. Learn about diagonalizable matrices, algebraic and geometric multiplicity, and eigenspaces. With this course, you will understand eigenvalues and eigenvectors completely.


January 11, 2021

Created by Dave

This course begins Summer 2021.
Current Status
Not Enrolled
Get Started

So you are in a math class, and your thinking: everything in this course is a goldmine. You connect with it, and all the pieces seem to fall together. Has this happened to you? Introducing Dave’s online math course: Eigenvalues and Eigenvectors. In this course, you will understand how everything in linear algebra fits together. This is a must-have course!

Who This Course is For

  • Students in college who want to learn more mathematics.
  • Anyone interested in eigenvalues and eigenvectors.
  • Students who are enrolled in linear algebra and want to improve their grade.
  • Anyone interested in majoring in mathematics, physics, or engineering.
  • Anyone wanting to learn about diagonalizable matrices, finding eigenvalues and eigenvectors, dynamical systems, and complex eigenvalues.


There is no required textbook, though you will need an up-to-date web browser, paper, and pen. A hand-held scientific calculator of your choice is recommended.

What You’ll Learn in Eigenvalues and Eigenvectors

  • Diagonalizable Matrices
  • Eigenvectors, Eigenvalues, and Eigenbases
  • Characterizations of Invertible Matrices
  • Dynamical Systems and Eigenvectors
  • Eigenvalues and Determinants
  • Characteristic Equation
  • Characteristic Polynomial
  • Algebraic Multiplicity of an Eigenvalue
  • Eigenvalues, Determinant, and Trace
  • Eigenspaces
  • Geometric Multiplicity
  • Eigenvalues and Similarity
  • Strategy for Diagonalization
  • Equilibria for Regular Transition Matrices
  • The Eigenvalues of a Linear Transformation
  • Complex Numbers
  • Fundamental Theorem of Algebra
  • Complex Eigenvalues and Eigenvectors
  • Trace, Determinant, and Eigenvalues
  • Stable Equilibrium
  • Dynamical Systems with Complex Eigenvalues

Course Description

We begin this course with a study of diagonalizable matrices. Simply put, a linear transformation T is diagonalizable if the matrix of T (with respect to some basis) is diagonal. Since we wish to diagonalize a matrix, we need a thorough understanding of eigenvalues and eigenvectors. After examining these concepts (including eigenbases) and the properties of eigenvectors, we work through several examples of diagonalization. At the end of the first lesson, we add additional characterizations of invertible matrices and begin exploring dynamical systems.

For the next two lessons, we focus on finding eigenvalues and finding eigenvectors, respectively. These studies lead us to the concepts of the algebraic multiplicity of an eigenvalue and the geometric multiplicity of an eigenvalue. This work culminates in a thorough understanding of eigenbases and a strategy for diagonalization.

In the next lesson, we develop more on dynamic systems, including the equilibria for regular transition matrices and several other topics. At the end of the course, we examine complex eigenvalues and eigenvectors.

Recommended Prerequisites for Eigenvalues and Eigenvectors

I recommend the prerequisite course Determinants. If you’re not sure if this course is for you, checkout the course contents below or find out more by checking out my free articles on linear algebra.

Course Content

Expand All

David A. Smith

Mathematics Educator

I sincerely believe that the potential in every student can be unlocked. As an accomplished and dedicated instructor, I have firm confidence that I can provide significant value to your studies.

I relish teaching and use it as a channel for my creativity as I seek to deliver real-life and relevant connections between the mathematics that I teach and my students. Promoting an active learning environment and excitement about mathematics is my dream come true.

David Smith (Dave) has a B.S. and M.S. in Mathematics and has enjoyed teaching precalculus, calculus, linear algebra, and number theory at both the junior college and university levels for over 20 years. David is the founder and CEO of Dave4Math.