Lesson 6.2 · 21 of 25 lessons
Singular value decomposition
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
SVD splits one matrix into a Vᵀ rotation, a Σ scaling, and a U rotation
- 02
First Vt rotates the input axes and x
- 03
Then S stretches perpendicular axes by 2 and 1
- 04
Finally U selects the output direction
A complex transform becomes rotate → scale → rotate
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
grid off
relative-grid on
basis on
coordinates on
0 1 #Vt
-1 0
2 0 #S
0 1
0 -1 #U
1 0
1, 1 #x@
// SVD splits one matrix into a `Vᵀ` rotation, a `Σ` scaling, and a `U` rotation
checkpoint Singular value decomposition · 1
// First {{Vt}} rotates the input axes and {{x}}
Vt
checkpoint Singular value decomposition · 2
// Then {{S}} stretches perpendicular axes by 2 and 1
S
checkpoint Singular value decomposition · 3
U
// Finally {{U}} selects the output direction\nA complex transform becomes **rotate → scale → rotate**
focus x03 · Unit 6
Lessons in this unit
Positive definite matrices & SVD — Read positive definiteness as energy and split SVD into rotate, scale, rotate.