Dave4Math / Courses / Jordan Canonical Forms

Jordan Canonical Forms

Master the Jordan Conical Form. Learn about the reduction to Jordan Canonical Form and the Algorithm for a Jordan Canonical Form. With this course, you'll have worked through several examples, and you'll be a pro at finding the Jordan Conical Form of a matrix.


January 26, 2021

Created by Dave

This course begins Summer 2021.
Current Status
Not Enrolled
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Have you ever seen a linear algebra book that didn’t even have the Jordan canonical form in it? Yes, they exist. Introducing Dave’s online math course: Jordan Canonical Forms. In this course, Dave reviews linear algebra’s most essential concepts and takes an in-depth look at the Jordan canonical form. Many examples are analyzed, and Dave discusses why this is a critical subject.

Who This Course is For

  • Students in college who want to learn more mathematics.
  • Anyone interested in Jordan canonical forms.
  • 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 the structure of a linear transformation and an algorithm for Jordan canonical form and Jordan basis.


There is no required textbook, though you will need an up-to-date web browser, paper, and pen. A hand-held scientific calculator is not necessary.

What You’ll Learn in Jordan Canonical Forms

  • Fundamentals on Vector Spaces and Linear Transformations
  • Bases and Coordinates
  • Linear Transformations and Matrices
  • Some Special Matrices and Polynomials
  • Subspaces, Complements, and Invariant Subspaces
  • The Structure of a Linear Transformation
  • Eigenvalues, Eigenvectors, and Generalized Eigenvectors
  • The Minimal Polynomial
  • Reduction to Normal Form
  • The Diagonalizable Case
  • Reduction to Jordan Canonical Form
  • The ESP of a Linear Transformation
  • The Algorithm for a Jordan Canonical Form
  • More Examples

Course Description

In this mostly self-contained course, we begin with a brisk review of the fundamentals of vector spaces and linear transformations. In the beginning, we focus on bases and coordinates and work through the details of linear transformations and matrices. Then we emphasize some special types of matrices that will play an important role later. We also discuss at length: polynomials, subspaces, complements, and invariant subspaces.

The goal in the next lesson is to understand the structure of a linear transformation. We begin by reviewing eigenvalues, eigenvectors, and generalized eigenvectors. Here all the theory is laid out for understanding the Jordan canonical form. This theory is an extended argument, so we carry it out in steps.

In the final lesson, we work through several examples to demonstrate an algorithm for finding Jordan canonical form and finding a Jordan basis.

Recommended Prerequisites for Jordan Canonical Forms

I recommend the prerequisite course Operators on Vector Spaces. 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

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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.