MAT 665 Home Page, Spring 2026

syllabus. Part of the syllabus are these resources: Department of Mathematics and Statistics (DoMS) policies and here is a pdf version of the NAU Syllabus Policy Statements.

homework.

Instructor information, including contact information. My office hours are
Monday 1:00-2:00,
Tuesday and Thursday 11:30-12:30, and
Friday 1:00-3:00.
Here is my weekly schedule. You can always send me e-mail, drop in, or make an appointment if these times aren't convenient.

Here are some graphical resources for differential equations:
Slope Field and Vector Field applet by Darryl Nester of Bluffton University.
Here is the OLD interface of that program, which has some advantages.

Vector Field applet written by Ariel Barton of the University of Arkansas.

Schedule in reverse Chronological Order

The Swift method can be done in generalized eigenspaces. Here is an example of a matlab file, swiftExample.mlx. Here is the solution.
Here is anonther example of a matlab file, swiftExample2.mlx, involving complex ("really complex") eigenvalues. Here is the solution.

Section 1.8: The Jordan Forms
Wikipedia page on the Jordan Canonical Form.

The Schaum Outline for matrix operations has some relevant material about Canonical Bases (pg. 82) and Jordan Canonical Form (pg. 91).

Here is my Algorithm for finding P such that inv(P)*A*P = B is in Jordan Canonical form: page 1, page 2, page 3. Here is a pdf of this same algorithm, but with a different labeling of the generalized eigenvectors. It uses the labeling in the book. Here is a set of worked examples from Perko, using that labeling.

Here is an example we started in class on Friday, March 27, continued on Monday, and hopefully finished on Wednesday. Here is a scan of the hand-written solutions of this example. Here is a MATLAB notebook ready to do this example, and here is the solution.
We will also do the JCF decomposition with this matlab file to find the general solution to the same ODE for which we did the S+N decomposition. The eigenvalues of A are 1,1,2.

For complex (and nonreal) eigenvalues, we ''complexify'' the ODE as described JCF complex 1 and JCF complex 2.

Here are some MATLAB files for examples in class:
repeated complex matrix no. 5 with the solution. This is very much like the homework assignment posted to CANVAS.

The swift method: Here is a handout about repeated complex eigenvalues, including the swift method.
repeated complex matrix no. 3 with the solution. This solution uses the Swift Method for a 4x4 matrix with \(\delta_1 = 2\), so it does not find the JCF of the matrix.
complex_6x6 asks you to solve the ODE for a \(6 \times 6\) matrix with repeated eigenvalues using the Swift method. Here is the solution.

Section 1.7: Multiple Eigenvalues Generalized Eigenvectors and the S+N decompostition
Multiplicty is by default the algebraic multiplicy of an eigenvalue. So an \(n \times n\) matrix has \(n\) eigenvalues, counting multiplicty. The geometric multiplicty is the maximum number of linearly independent eigenvectors.
Here is the wikipedia page on Generalized Eigenvectors.
You can search for "semisimple plus nilpotent decomposition of a real matrix" and get some information, but I cannot find a good web page describing this result. The most useful application of the S+N decompositon follows from section 1.7, Theorem 1:

Theorem Suppose \(A \in \mathbb{R}^{n \times n}\) has a single eigenvalue \(\lambda\) with multiplicty \(n\). The S+N decomposition has \(S = \lambda I_n\), and \(N = A - \lambda I_n\) is nilpotent of order at most \(n\). The solution to \(\dot{\bf x} = A {\bf x}, \ {\bf x}(0) = {\bf x}_0\) is \({\bf x}(t) = e^{\lambda t} \left (I_n + N t + N^2 \frac{t^2}{2} + N^3 \frac{t^3}{3!} + \cdots + N^{n-1} \frac{t^{n-1}}{(n-1)!} \right ) {\bf x}_0 \). If \({\bf x}_0\) is a given vector, then the solution is more efficiently found first by computing \({\bf x}_{i+1} = N {\bf x}_{i}\) for \(0 \leq i \leq n-2\). Then the solution to the IVP is \({\bf x}(t) = e^{\lambda t} \left( {\bf x}_0 + t {\bf x}_1 + \frac{t^2}{2} {\bf x}_2 + \frac{t^3}{3!} {\bf x}_3 + \cdots +\frac{t^{n-1}}{(n-1)!}{\bf x}_{n-1} \right) \).

Here is a scanned solution of Perko's problem 1.7 problem 2b, where the eigenvalues are -1,-1,1. Note the moral at the end of the file!
Here is a MATLAB program that computes the S+N with symbolic matlab. when the eigenvalues are 1, 1, 2.

Computer Algebra
In this section and beyond, you can use mathematica (or matlab, or some other program) to do symbolic (not floating point) matrix operations with Mathematica or Matlab or some other program.
The allowed operations are differentiation, matrix multiplication, finding eigenvalues, computing inverses, and computing the reduced row echelon form of a matrix.
For example here is a Mathematica notebook.
We will mostly use MATLAB because the syntax is simpler. An example of MATLAB code is in the notes for section 1.7.

Section 1.6: Complex Eigenvalues
The goal is to understand the ODE for 2x2 matrices with complex eigenvalues:
Find the general solution in a few different ways.
Draw accurate phase portraits.
Solve the initial value problem using the Swift method.
Calculate exp(At) using the Swifter method.

This complex eigenvalues pdf describes the general solution and describes how to draw the phase portraits.
Feb 27 Here is a scan of a worked example with eigenvalues -1 +/- 2i, similar to an example in the pdf. For this same ODE, here is a Mathematic file that plots the phase portrait.

Feb 27 Here is a pdf describing the Swift Method for MAT 239 students.

Feb 27 Here is a worksheet I gave to MAT 239 students: worksheet 29.5. Check out example 3 in Paul's notes on Complex Eigenvalues, where does this same problem with the slow method.

Here is a draft of a more advanced paper entitled Shortcuts for solving some initial value problems.

Section 1.3:
Desmos graph of \(\{ A {\bf x} : |{\bf x}| = 1 \}\) for a \(2\times 2\) matrix \(A\).

Here is a proof of Corollary 4 that is different from the book.

Section 1.2
The general solution to \(\dot{\bf x} = A {\bf x}\), where \(A\) is a \(2 \times 2\) real matrix with real eigenvalues \(\lambda_1 \neq \lambda_2\) and associated eigenvectors \({\bf v}_1\) and \({\bf v}_2\) is \[{\bf x}(t) = c_1 \exp(\lambda_1 t) {\bf v}_1 + c_2 \exp(\lambda_2 t) {\bf v}_2.\] This is the most useful way to write the general solution when sketchinng the phase portrait, and no matrix inverse is required! This general solution can also be written as \[{\bf x}(t) = P \, \text{diag}[\exp(\lambda_1 t), \exp(\lambda_2 t)] {\bf c} = [ \exp(\lambda_1 t) {\bf v}_1 \ \exp(\lambda_2 t) {\bf v}_2 ] {\bf c},\] where \(P = [ {\bf v}_1 \ {\bf v}_2 ]\). If you want to solve the ODE with a specific initial condition \({\bf x}(0) = {\bf x}_0\) you can solve \(P {\bf c} = {\bf x}_0\) for \({\bf c}\) by Gaussian elimination without inverting \(P\). (Admittedly, this is mostly useful for \(3 \times 3\) or larger matrices.) To solve the initial value problem with an arbitrary initial condition, then \({\bf c} = P^{-1} {\bf x}(0)\), and the solution can be written as \[{\bf x}(t) = P \, \text{diag}[\exp(\lambda_1 t), \exp(\lambda_2 t)] P^{-1}{\bf x}(0).\] (This is the Corollary in section 1.2 of the book.)

Section 1.1
Here are trajectories (without arrows) for dx/dt = x, dy/dt = ay
Here are some examples of three-dimensional phase portraits of uncoupled linear systems in \(\mathbb{R}^3\). I notice that Perko’s homework on section 1.1 doesn’t need this type of phase portrait, so I added a bonus problem to homework 2.

On January 26, I discussed the general solution to \(\frac{dy}{dx} = y^2\), which is an expression \(\phi(x, c)\) that gives a solution to the ODE for every fixed \(c\). \(\phi: \Omega \to \mathbb R\), where \(\Omega \subseteq \mathbb R \times \mathbb R\), defined by \(\phi(x, 0) =0\) for all \(x\), and \[ \phi(x, c) = \frac{1}{\ln|c| -x} \text{ on the interval }\{x\in \mathbb{R} :\text{sgn}(c)(\ln|c| -x) > 0 \} \text{ if } c \neq 0. \] Note: This includes the solution with \(y(-1) = 1\), when \(c = 1\). See the desmos graph y' = y^2 general solution.
I have made another desmos graph, called y' = y^2, alternative general solution, which has some clues about how to do Bonus 3. It derives a different general solution: \[ \phi(x, c) = \frac{\tanh^{-1}(c)}{1 - \tanh^{-1}(c) \, x} \text{ on the interval }\{x\in \mathbb{R} :1-\tanh^{-1}(c) \, x > 0 \} \text{ if } -1 < c < 1,\] \[ \phi(x, c) = \frac{1}{ 1 + c - x} \text{ on } (-\infty, c+1) \text{ if } c \leq -1,\] \[ \phi(x, c) = \frac{1}{ -1 + c - x} \text{ on } (c-1, \infty) \text{ if } c \geq 1.\]

Review of Undergraduate Differential Equations
Formula Sheet from a past MAT 239 (Differential Equations) Final Exam.
Theory of Linear Homogeneous ODEs from a previous MAT 239 class.
239 Final Exam from a previous semester. Here are the scanned solutions.