Graduate · Coercivity in a finite model · 549 of 650
Coercivity in a finite model · First diagonal entry a=0.6; Second diagonal entry b=1.6
Use θ=6π/5 radians. Find vᵀAv and the sharp lower and upper bounds over all unit directions. Givens: First diagonal entry a=0.6; Second diagonal entry b=1.6.
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(0.6,1.6) 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.
Use this scratch space or work on paper. Your notes stay on this page and clear when you leave. Answers are for self-checking; they are not automatically graded.
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(0.6,1.6) 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=(0.6)cos²θ+(1.6)sin²θ=0.945492. Since cos²θ+sin²θ=1, this weighted average lies between 0.6 and 1.6.
Step 3
Check the mathematical meaning
Since cos²θ+sin²θ=1, 0.6≤0.945492≤1.6. The lower bound is attained in an eigenvector direction and is strictly positive. Direction radians: 3.77; Quadratic form value: 0.945; Coercivity lower bound: 0.6; Upper bound: 1.6. Decimal values are rounded, so use unrounded intermediate values.
The answer
At θ=3.769911, vᵀAv=(0.6)cos²θ+(1.6)sin²θ=0.945492. Since cos²θ+sin²θ=1, this weighted average lies between 0.6 and 1.6. Since cos²θ+sin²θ=1, 0.6≤0.945492≤1.6. The lower bound is attained in an eigenvector direction and is strictly positive. Animation check: Direction radians: 3.77; Quadratic form value: 0.945; Coercivity lower bound: 0.6; Upper bound: 1.6. 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.
Use Play, Pause, and the timeline to inspect the construction. Reset restores the question’s original settings. The displayed assumptions describe where this model applies.
Watch the relationship
PausedHD animation studio
From experiment to screen.
Present a crisp Canvas scene, save a full-HD image, or capture your model as a silent video.
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(0.6,1.6) 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.