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Teaching video
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Undergraduate chapters and video availability01 · Read and understand
What you will learn
- Compute eigenvalues and eigenvectors of a symmetric coupling matrix.
- Justify the conclusion "A(v+w)=3v+w" using the stated assumptions.
Before you start
Determinants and linear systems.
Keep paper nearby. Read the question once for the context, then again to identify what is known and what you need to find.
Start with a question
Find the eigenmodes of A=[[2,1],[1,2]].
Why this math matters
Compute eigenvalues and eigenvectors of a symmetric coupling matrix. This worked micro-lesson connects a precise mathematical condition to a conclusion you can check. The transfer task asks you to change the setting and decide which parts of the reasoning still apply.
Set up the model
A useful answer starts with clear assumptions:
- The matrix acts on R².
- Eigenmodes describe this linear model only.
02 · Work through the example
Follow the reasoning, one step at a time.
Try to predict the next step before reading it. After each calculation, explain why the operation makes sense and how it helps answer the original question.
Work through it with Amar
See this example unfold.
The complete worked example, one idea at a time.
Find directions a coupled system preserves
PausedQuestion: Start with the question. Paused.
Question
Start with the question
Find the eigenmodes of A=[[2,1],[1,2]].
Before you calculate
Read what is known and what you need to find. Make a prediction before moving to the first calculation.
Starts paused. Play advances through the full text at a reading pace; pause whenever you need more time. Previous, Next, and the phase buttons let you set your own pace. Playback pauses when this walkthrough leaves the screen or you switch tabs.
Your device’s reduced-motion setting keeps each phase still. Manual controls remain available. The full written solution stays below.
Build the model
det(A−λI)=(2−λ)²−1=(λ−1)(λ−3)
Eigenvalues make the shifted matrix singular.
Work through the mathematics
λ=3 has v=(1,1); λ=1 has w=(1,−1)
Substitution identifies common and opposing modes.
Check the conclusion
A(v+w)=3v+w
Any input decomposed into these two modes evolves by independent scaling factors.
The result
A(v+w)=3v+w
Any input decomposed into these two modes evolves by independent scaling factors.
Common mistakes to catch
- Eigenvalues scale vectors; they are not vector coordinates.
- A repeated eigenvalue does not automatically supply enough independent eigenvectors.
03 · Practice independently
Try it before revealing the answer.
Use paper or a calculator as needed. Write your units and reasoning, then open the hint or explanation to check your approach.
Practice 1
What is A²v?
Show a hint
Apply the eigenvalue twice.
Reveal answer and explanation
9v
A²v=A(3v)=3Av=9v.
Practice 2
Can the zero vector identify an eigenvalue?
Show a hint
Check Av=λv when v=0.
Reveal answer and explanation
No
The equality holds for every λ when v=0, so eigenvectors must be nonzero.
Take the idea with you
Interpret common-mode and difference-mode behavior in two coupled measurements.
04 · Reflect and continue
Can you explain it in your own words?
Before moving on, explain the main idea without looking at the worked example. Try both practice questions, check your reasoning, and name one mistake you now know how to avoid. Return to a step if you still need support.
Up next: Compute a large matrix power through its eigenbasis
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