Graduate · Coercivity in a finite model · 589 of 650
Coercivity in a finite model · First diagonal entry a=2.4; Second diagonal entry b=0.4
Use θ=6π/5 radians. Find vᵀAv and the sharp lower and upper bounds over all unit directions. Givens: First diagonal entry a=2.4; Second diagonal entry b=0.4.
Try the question, use a hint when you need one, and compare your reasoning with the worked solution. The animation starts with this chapter’s values; changing its controls explores a new case.
01 · Make a prediction
Your practice question
Let A=diag(2.4,0.4) and v=(cos θ,sin θ), so |v|=1. Use θ=6π/5 radians. Find vᵀAv and the sharp lower and upper bounds over all unit directions.
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A starting point
The form is a cos²θ+b sin²θ, a weighted average of the two positive diagonal entries.
Work through the reasoning
Step 1
Identify the model and target
Let A=diag(2.4,0.4) and v=(cos θ,sin θ), so |v|=1. Use θ=6π/5 radians. The governing relation is vᵀAv=a cos²θ+b sin²θ≥min(a,b)|v|²; |v|=1. The form is a cos²θ+b sin²θ, a weighted average of the two positive diagonal entries.
Step 2
Substitute and calculate
At θ=3.769911, vᵀAv=(2.4)cos²θ+(0.4)sin²θ=1.709017. Since cos²θ+sin²θ=1, this weighted average lies between 0.4 and 2.4.
Step 3
Check the mathematical meaning
Since cos²θ+sin²θ=1, 0.4≤1.709017≤2.4. The lower bound is attained in an eigenvector direction and is strictly positive. Direction radians: 3.77; Quadratic form value: 1.709; Coercivity lower bound: 0.4; Upper bound: 2.4. Decimal values are rounded, so use unrounded intermediate values.
The answer
At θ=3.769911, vᵀAv=(2.4)cos²θ+(0.4)sin²θ=1.709017. Since cos²θ+sin²θ=1, this weighted average lies between 0.4 and 2.4. Since cos²θ+sin²θ=1, 0.4≤1.709017≤2.4. The lower bound is attained in an eigenvector direction and is strictly positive. Animation check: Direction radians: 3.77; Quadratic form value: 1.709; Coercivity lower bound: 0.4; Upper bound: 2.4. Decimal displays are rounded; retain the original parameters and exact π until the final step.
Compare the method as well as the result. A different valid method may reach the same answer. Keep exact values until the last step when the question asks for rounding.
02 · Explore the model
See the mathematical relationship move.
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Watch the relationship
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The mathematical idea
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. Let A=diag(2.4,0.4) and v=(cos θ,sin θ), so |v|=1. Use θ=6π/5 radians. The requested state occurs at 60% playback; use the exact target stated in the question for your calculation.
vᵀAv=a cos²θ+b sin²θ≥min(a,b)|v|²; |v|=1
03 · Reflect and transfer
Explain what changes and why.
Why does this finite-dimensional positive diagonal example not by itself establish coercivity for a different infinite-dimensional operator?
This is a distinct guided scenario using a reusable mathematical model. Similar-looking diagrams can represent different given values and conclusions; they are not different mathematical theories.