Lesson 6.1 · 20 of 25 lessons
Positive definiteness
Read positive definiteness as energy and split SVD into rotate, scale, rotate.
Open the interactive animation01 · Story
What to verify in this scene
Follow the explanations and operations, then identify the mathematical evidence that remains in the final scene.
- 01
A symmetric A is positive definite when xᵀAx is positive in every nonzero direction
- 02
Keep x at its original position and apply A to a copied input to create a separate result Ax
- 03
With both x and Ax visible, take their dot product to measure the energy in this direction
- 04
dot(x, Ax) = 14 > 0
This calculation is one directional example - 05
Now let u = (a,b) be any nonzero vector and check every direction at once
- 06
uᵀAu = 2a² + 2ab + 2b² = (a+b)² + a² + b² > 0
Therefore A is positive definite in every nonzero direction
02 · Notebook
The runnable notebook
This source creates the mathematical objects and controls the order of operations and animation. Read it here, then play the same scene yourself.
2d
zoom 4/5
coordinates on
2 1 #A
1 2
// A symmetric {{A}} is **positive definite** when `xᵀAx` is positive in every nonzero direction
checkpoint Positive definiteness · 1
1, 2 #x@
// Keep {{x}} at its original position and apply {{A}} to a copied input to create a separate result `Ax`
A * x #Ax@
// With both {{x}} and {{Ax}} visible, take their dot product to measure the energy in this direction
dot(x, Ax) #energy@
// `dot({{x}}, {{Ax}}) = 14 > 0`\nThis calculation is **one directional example**
focus x Ax
checkpoint Positive definiteness · 2
focus -
clear
field board
A +
// Now let `u = (a,b)` be any nonzero vector and check every direction at once
// `uᵀAu = 2a² + 2ab + 2b² = (a+b)² + a² + b² > 0`\nTherefore {{A}} is **positive definite** in every nonzero direction03 · Unit 6
Lessons in this unit
Positive definite matrices & SVD — Read positive definiteness as energy and split SVD into rotate, scale, rotate.