Lab 24: Linear Algebra and Differential Equations

Lab goals

In this independent-study lab, you will use Python and an interactive visualization to study linear differential equations through linear algebra.

By the end, you should be able to:

solve $\mathbf x'=A\mathbf x$ using $e^{At}$;
compute eigenvalues and use them to classify stability;
compare exact matrix exponential solutions with forward Euler approximations;
simulate diffusion on a graph using a graph Laplacian;
recognize how eigenvectors are the natural coordinates for differential equations.

Python practice notebook

Download and work through the notebook first:

The notebook contains worked examples, similar practice questions, and solutions. It is designed for independent study.

Interactive lab

After the notebook, use the interactive lab:

Open Lab 24 interactive HTML

The interactive page lets you explore:

a two-dimensional system $\mathbf x'=A\mathbf x$;
eigenvalues, trace, determinant, and stability;
vector fields and trajectories;
forward Euler time stepping;
graph heat diffusion on a path graph.

Independent-study instructions

For each section, do the following:

Read the short explanation.
Run the Python code.
Answer the practice question before opening the solution.
Use the interactive page to test at least two additional examples.
Write a short reflection connecting the computation to the theorem.

Problems to submit or discuss

Problem 1: Matrix exponential solution

Let

\[ A=\begin{bmatrix}2&1\\1&2\end{bmatrix}, \qquad \mathbf x_0=\begin{bmatrix}4\\2\end{bmatrix}. \]

Use eigenvectors to find $\mathbf x(t)$.

Solution

The eigenpairs are

\[ \lambda_1=3, \quad \mathbf v_1=\begin{bmatrix}1\\1\end{bmatrix}, \qquad \lambda_2=1, \quad \mathbf v_2=\begin{bmatrix}1\\-1\end{bmatrix}. \]

Since

\[ \begin{bmatrix}4\\2\end{bmatrix} =3\begin{bmatrix}1\\1\end{bmatrix} +1\begin{bmatrix}1\\-1\end{bmatrix}, \]

we have

\[ \mathbf x(t)=3e^{3t}\begin{bmatrix}1\\1\end{bmatrix} +e^t\begin{bmatrix}1\\-1\end{bmatrix}. \]

Problem 2: Stability

Classify the equilibrium $\mathbf 0$ for

\[ A=\begin{bmatrix}-1&-4\\1&-1\end{bmatrix}. \]

Solution

The characteristic polynomial is

\[ \lambda^2+2\lambda+5=0. \]

Thus

\[ \lambda=-1\pm2i. \]

The real parts are negative, so the equilibrium is asymptotically stable. The nonzero imaginary parts mean solutions spiral as they decay.

Problem 3: Jordan block

Compute $e^{Jt}$ for

\[ J=\begin{bmatrix}3&1\\0&3\end{bmatrix}. \]

Solution

Write $J=3I+N$, where

\[ N=\begin{bmatrix}0&1\\0&0\end{bmatrix} \]

and $N^2=0$. Therefore

\[ e^{Jt}=e^{3t}(I+tN) =e^{3t}\begin{bmatrix}1&t\\0&1\end{bmatrix}. \]

Problem 4: Euler stability

For $x'=-10x$, determine the range of $h>0$ for which forward Euler decays.

Solution

Forward Euler gives

\[ x_{k+1}=(1-10h)x_k. \]

Decay requires

\[ |1-10h|<1. \]

Thus

\[ 0<h<\frac15. \]

Reflection questions

Why does an eigenbasis make a system easier to solve?
What does the sign of $\operatorname{Re}(\lambda)$ tell you about long-time behavior?
Why can Euler’s method be unstable even when the exact system is stable?
In the graph heat equation, why does the final state become constant on a connected graph?

AI companion prompts

Try these prompts after completing your own work:

“Explain the matrix exponential solution of $\mathbf x'=A\mathbf x$ using a $2\times2$ example.”
“Give me a stable spiral matrix and explain its eigenvalues.”
“Help me compare forward Euler with the exact matrix exponential for a stiff system.”
“Explain graph heat diffusion using eigenvectors of the Laplacian.”

# Lab 24: Linear Algebra and Differential Equations {.unnumbered} # Lab goals In this independent-study lab, you will use Python and an interactive visualization to study linear differential equations through linear algebra. By the end, you should be able to: 1. solve $\mathbf x'=A\mathbf x$ using $e^{At}$; 2. compute eigenvalues and use them to classify stability; 3. compare exact matrix exponential solutions with forward Euler approximations; 4. simulate diffusion on a graph using a graph Laplacian; 5. recognize how eigenvectors are the natural coordinates for differential equations. # Python practice notebook Download and work through the notebook first: - [Download Lab 24 Python notebook](lab-24-linear-algebra-differential-equations-independent-study.ipynb) - [Open Lab 24 in Google Colab](https://colab.research.google.com/github/wanghemath/MATH5110/blob/main/Labs/lab-24-linear-algebra-differential-equations-independent-study.ipynb) The notebook contains worked examples, similar practice questions, and solutions. It is designed for independent study. # Interactive lab After the notebook, use the interactive lab: [Open Lab 24 interactive HTML](lab-24-interactive.html) The interactive page lets you explore: - a two-dimensional system $\mathbf x'=A\mathbf x$; - eigenvalues, trace, determinant, and stability; - vector fields and trajectories; - forward Euler time stepping; - graph heat diffusion on a path graph. # Independent-study instructions For each section, do the following: 1. Read the short explanation. 2. Run the Python code. 3. Answer the practice question before opening the solution. 4. Use the interactive page to test at least two additional examples. 5. Write a short reflection connecting the computation to the theorem. # Problems to submit or discuss ## Problem 1: Matrix exponential solution Let $$ A=\begin{bmatrix}2&1\\1&2\end{bmatrix}, \qquad \mathbf x_0=\begin{bmatrix}4\\2\end{bmatrix}. $$ Use eigenvectors to find $\mathbf x(t)$. <details> <summary>Solution</summary> The eigenpairs are $$ \lambda_1=3, \quad \mathbf v_1=\begin{bmatrix}1\\1\end{bmatrix}, \qquad \lambda_2=1, \quad \mathbf v_2=\begin{bmatrix}1\\-1\end{bmatrix}. $$ Since $$ \begin{bmatrix}4\\2\end{bmatrix} =3\begin{bmatrix}1\\1\end{bmatrix} +1\begin{bmatrix}1\\-1\end{bmatrix}, $$ we have $$ \mathbf x(t)=3e^{3t}\begin{bmatrix}1\\1\end{bmatrix} +e^t\begin{bmatrix}1\\-1\end{bmatrix}. $$ </details> ## Problem 2: Stability Classify the equilibrium $\mathbf 0$ for $$ A=\begin{bmatrix}-1&-4\\1&-1\end{bmatrix}. $$ <details> <summary>Solution</summary> The characteristic polynomial is $$ \lambda^2+2\lambda+5=0. $$ Thus $$ \lambda=-1\pm2i. $$ The real parts are negative, so the equilibrium is asymptotically stable. The nonzero imaginary parts mean solutions spiral as they decay. </details> ## Problem 3: Jordan block Compute $e^{Jt}$ for $$ J=\begin{bmatrix}3&1\\0&3\end{bmatrix}. $$ <details> <summary>Solution</summary> Write $J=3I+N$, where $$ N=\begin{bmatrix}0&1\\0&0\end{bmatrix} $$ and $N^2=0$. Therefore $$ e^{Jt}=e^{3t}(I+tN) =e^{3t}\begin{bmatrix}1&t\\0&1\end{bmatrix}. $$ </details> ## Problem 4: Euler stability For $x'=-10x$, determine the range of $h>0$ for which forward Euler decays. <details> <summary>Solution</summary> Forward Euler gives $$ x_{k+1}=(1-10h)x_k. $$ Decay requires $$ |1-10h|<1. $$ Thus $$ 0<h<\frac15. $$ </details> # Reflection questions 1. Why does an eigenbasis make a system easier to solve? 2. What does the sign of $\operatorname{Re}(\lambda)$ tell you about long-time behavior? 3. Why can Euler's method be unstable even when the exact system is stable? 4. In the graph heat equation, why does the final state become constant on a connected graph? # AI companion prompts Try these prompts after completing your own work: - “Explain the matrix exponential solution of $\mathbf x'=A\mathbf x$ using a $2\times2$ example.” - “Give me a stable spiral matrix and explain its eigenvalues.” - “Help me compare forward Euler with the exact matrix exponential for a stiff system.” - “Explain graph heat diffusion using eigenvectors of the Laplacian.”