Learn with Amar
Teaching video
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Undergraduate chapters and video availability01 · Read and understand
What you will learn
- Complete squares to test positive definiteness.
- Justify the conclusion "A=[[2,1],[1,2]] is positive definite" using the stated assumptions.
Before you start
Quadratic expressions and symmetric matrices.
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
Show that q(x,y)=2x²+2xy+2y² is positive for every nonzero pair.
Why this math matters
Complete squares to test positive definiteness. 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 is real and symmetric.
- Strict positivity is required for every nonzero vector.
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.
Certify that a quadratic energy is strictly positive
PausedQuestion: Start with the question. Paused.
Question
Start with the question
Show that q(x,y)=2x²+2xy+2y² is positive for every nonzero pair.
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
q=(x+y)²+x²+y²
Completing squares exposes nonnegative pieces.
Work through the mathematics
q=0 forces x=y=0
The last two squares vanish only at the origin.
Check the conclusion
A=[[2,1],[1,2]] is positive definite
Since q=zᵀAz, the positive quadratic form certifies a unique minimum at the origin.
The result
A=[[2,1],[1,2]] is positive definite
Since q=zᵀAz, the positive quadratic form certifies a unique minimum at the origin.
Common mistakes to catch
- Positive diagonal entries alone do not guarantee positive definiteness.
- Semidefinite permits nonzero vectors with zero quadratic value.
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
Is x²−y² positive definite?
Show a hint
Evaluate at (0,1).
Reveal answer and explanation
No
The value is −1, so the form is indefinite.
Practice 2
Is x² positive definite on R²?
Show a hint
Test a nonzero vector with x=0.
Reveal answer and explanation
No; it is positive semidefinite
The form vanishes along the entire y-axis.
Take the idea with you
Use a quadratic form to test whether a proposed energy detects every displacement direction.
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: Separate a small residual from a trustworthy solution
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