Graduate · Coercivity in a finite model
Coercivity in a finite model: An isotropic unit form
Coercivity in a finite model: investigate an isotropic unit form with first diagonal entry a = 1; second diagonal entry b = 1.
Predict what will happen, press Play, then pause and explain what changed. Every control also works without playback.
Watch the relationship
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Understand what you are seeing
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A coercivity bound controls a norm uniformly over all directions. In this positive diagonal finite-dimensional model, the smallest eigenvalue supplies the sharp bound. Sampling directions illustrates that bound, while the algebraic expression proves it for directions the animation never samples. This investigation starts with First diagonal entry a = 1; Second diagonal entry b = 1. Predict the result before playing, then change one parameter while keeping the other fixed to test the reason for the change.
A relationship to keep
vᵀAv=a cos²θ+b sin²θ≥min(a,b)|v|²; |v|=1
Read the symbols alongside the explanation. A diagram shows the relationship; the assumptions tell you when it applies.
- STEP 1
Set up the mathematical model
Choose positive diagonal entries and identify the proposed lower bound min(a,b). The starting case is “An isotropic unit form.”
- STEP 2
Follow the changing quantity
Rotate a unit vector and compare its quadratic energy with the lower and upper horizontal bounds.
- STEP 3
Explain and test the result
Rewrite the energy as a weighted average of a and b. Explain why positive diagonal entries suffice here but not for an arbitrary nondiagonal matrix.
Your turn to explain
Make a prediction. Test your reasoning.
Keep First diagonal entry a = 1; Second diagonal entry b = 1. Pause the timeline at 20%. Given direction radians = 1.257, calculate quadratic form value, coercivity lower bound, upper bound. Show the substitution into the displayed formula.
Use the values specified in the question. Reset restores the initial values for this investigation.
Compare your explanation
At θ=1.256637, vᵀAv=(1)cos²θ+(1)sin²θ=1. Since cos²θ+sin²θ=1, this weighted average lies between 1 and 1. Results: Quadratic form value: 1; Coercivity lower bound: 1; Upper bound: 1. Decimal values are rounded; retain the original parameters when checking.
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Connect the animation to a worked example and practice questions.