Learn with Amar
Teaching video
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Undergraduate chapters and video availability01 · Read and understand
What you will learn
- Build a graph Laplacian and connect its quadratic form to edge differences.
- Justify the conclusion "xᵀLx=4+1=5" using the stated assumptions.
Before you start
Matrices and finite graphs.
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 the path 1—2—3, compute xᵀLx at x=(1,3,2).
Why this math matters
Build a graph Laplacian and connect its quadratic form to edge differences. 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 graph is undirected and unweighted.
- Each edge is counted once in the energy sum.
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.
Turn a network into an energy matrix
PausedQuestion: Start with the question. Paused.
Question
Start with the question
For the path 1—2—3, compute xᵀLx at x=(1,3,2).
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
L=[[1,−1,0],[−1,2,−1],[0,−1,1]]
Put vertex degrees on the diagonal and minus one on adjacent off-diagonal positions.
Work through the mathematics
xᵀLx=(x₁−x₂)²+(x₂−x₃)²
Each undirected edge contributes one squared difference.
Check the conclusion
xᵀLx=4+1=5
Constant values have zero energy; this nonconstant signal has positive disagreement across the connected graph.
The result
xᵀLx=4+1=5
Constant values have zero energy; this nonconstant signal has positive disagreement across the connected graph.
Common mistakes to catch
- The Laplacian is degree minus adjacency, not the reverse here.
- Disconnected graphs have more than one independent zero-energy signal.
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
Find L(1,1,1).
Show a hint
Each row sums to zero.
Reveal answer and explanation
(0,0,0)
A constant signal has no edge differences.
Practice 2
If one edge is removed, what happens to the nullspace dimension?
Show a hint
Count connected components.
Reveal answer and explanation
It becomes two
Independent constants may be assigned to each of the two components.
Take the idea with you
Interpret the quadratic form as disagreement between neighboring temperature sensors.
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: Locate an optimum among feasible polygon corners
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