Lesson 5.2 · 19 of 25 lessons
Diagonalization
Start with directions preserved by a transform and see how diagonalization simplifies the action.
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
The columns of P are eigenvectors whose directions are preserved by A
- 02
In the eigenbasis, D only scales the coordinate axes independently
- 03
Following PD by Pinv produces A2
A = PDP⁻¹ is change basis, scale, and return
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.
field board
2 1 #A
1 2
1 1 #P
1 -1
3 0 #D
0 1
1/2 1/2 #Pinv
1/2 -1/2
// The columns of {{P}} are eigenvectors whose directions are preserved by {{A}}
checkpoint Diagonalization
// In the eigenbasis, {{D}} only scales the coordinate axes independently
P * D #PD@
PD * Pinv #A2@
// Following {{PD}} by {{Pinv}} produces {{A2}}\n`A = PDP⁻¹` is **change basis, scale, and return**03 · Unit 5
Lessons in this unit
Eigenvalues & eigenvectors — Start with directions preserved by a transform and see how diagonalization simplifies the action.