Learn with Amar
Teaching video
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Undergraduate chapters and video availability01 · Read and understand
What you will learn
- Read singular values from AᵀA and interpret a rank-one approximation.
- Justify the conclusion "A₁=[[3,0],[0,0],[0,0]]; ||A−A₁||₂=1" using the stated assumptions.
Before you start
Eigenvalues, norms, and matrix transpose.
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=[[3,0],[0,1],[0,0]], find its singular values and best rank-one truncation.
Why this math matters
Read singular values from AᵀA and interpret a rank-one approximation. 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 error norm with subscript 2 is the operator norm.
- The requested approximation has rank at most one.
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.
Keep the dominant direction of a rectangular map
PausedQuestion: Start with the question. Paused.
Question
Start with the question
For A=[[3,0],[0,1],[0,0]], find its singular values and best rank-one truncation.
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
AᵀA=diag(9,1)
Singular values are square roots of the nonnegative eigenvalues of this matrix.
Work through the mathematics
σ₁=3, σ₂=1; retain the first coordinate direction
The largest singular value identifies the greatest unit-input stretching.
Check the conclusion
A₁=[[3,0],[0,0],[0,0]]; ||A−A₁||₂=1
Removing the weaker orthogonal mode leaves spectral-norm error equal to its singular value.
The result
A₁=[[3,0],[0,0],[0,0]]; ||A−A₁||₂=1
Removing the weaker orthogonal mode leaves spectral-norm error equal to its singular value.
Common mistakes to catch
- Singular values are not eigenvalues of a rectangular matrix.
- A transpose and an inverse are different operations.
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 the Frobenius error here?
Show a hint
Only one nonzero discarded entry remains.
Reveal answer and explanation
1
The sum of squared discarded entries is one.
Practice 2
Can a singular value be negative?
Show a hint
It is a square root of an eigenvalue of AᵀA.
Reveal answer and explanation
No
vᵀAᵀAv=||Av||² is nonnegative.
Take the idea with you
Explain how discarding small singular directions compresses a linear data model.
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: Certify that a quadratic energy is strictly positive
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