Learn with Amar
Teaching video
A teaching recording for this chapter has not been published yet. Start with the worked example below and explore the related animations where available.
Graduate chapters and video availability01 · Read and understand
What you will learn
- Use a generalized eigenvector when an eigenvalue lacks a full eigenbasis.
- Justify the conclusion "Aⁿ=2ⁿI+n2ⁿ⁻¹N for n≥1" using the stated assumptions.
Before you start
Eigenvalues and matrix powers.
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
For A=[[2,1],[0,2]], explain why diagonalization fails and compute Aⁿ.
Why this math matters
Use a generalized eigenvector when an eigenvalue lacks a full eigenbasis. 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:
- n is a positive integer in the displayed power formula.
- The scalar field has characteristic zero.
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.
Recover a missing eigenvector with a Jordan chain
PausedQuestion: Start with the question. Paused.
Question
Start with the question
For A=[[2,1],[0,2]], explain why diagonalization fails and compute Aⁿ.
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
A=2I+N, where N=[[0,1],[0,0]] and N²=0
The nilpotent part records the coupling missed by the repeated eigenvalue alone.
Work through the mathematics
(A−2I)e₂=e₁; (A−2I)e₁=0
These vectors form a length-two generalized eigenvector chain.
Check the conclusion
Aⁿ=2ⁿI+n2ⁿ⁻¹N for n≥1
All higher binomial terms vanish because N²=0; the extra polynomial factor distinguishes Jordan growth from pure diagonal growth.
The result
Aⁿ=2ⁿI+n2ⁿ⁻¹N for n≥1
All higher binomial terms vanish because N²=0; the extra polynomial factor distinguishes Jordan growth from pure diagonal growth.
Common mistakes to catch
- A repeated eigenvalue does not imply the matrix is a scalar matrix.
- Generalized eigenvectors satisfy a chain relation, not the ordinary eigenvector equation.
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
Compute A³.
Show a hint
Substitute n=3.
Reveal answer and explanation
[[8,12],[0,8]]
The off-diagonal term is 3·2²=12.
Practice 2
How many independent ordinary eigenvectors exist?
Show a hint
Solve Nv=0.
Reveal answer and explanation
One direction, span(e₁)
The second coordinate must vanish, so no two-vector eigenbasis exists.
Take the idea with you
Compare polynomial-times-exponential growth with pure exponential growth in a defective linear system.
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: Find the shortest polynomial that annihilates a matrix
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