Linear Algebra

Singular Value Decomposition

Every matrix is three clean steps: rotate, stretch along the axes, rotate again. Watch the unit circle become an ellipse one step at a time, then use the same idea to compress a picture.

v₁ → σ₁u₁v₂ → σ₂u₂unit circle and its imagethe letter F, carried along

Singular values σ₁, σ₂—

—

View

Matrix A

1.200.800.301.10

a 1.20

b 0.80

c 0.30

d 1.10

Stage of A = UΣVᵀ whole

circleVᵀ · Σ · U

Presets

condition σ₁/σ₂—

det A = ±σ₁σ₂—

rank—

What to observe

Animate the stages: Vᵀ turns v₁, v₂ onto the axes, Σ stretches along the axes by σ₁ and σ₂, U turns them onto u₁, u₂. The result is exactly the ellipse A makes of the circle.
On Flip, det A < 0 and the F comes out mirrored. Two rotations and a stretch cannot do that, so one of U, V must be a reflection: here U flips the plane over before it turns.
On Rank 1, σ₂ = 0: the Σ step squashes the circle and the F onto a segment. A whole direction is lost, σ₁/σ₂ is infinite and A cannot be inverted.
Compress the house: keeping the first 8 terms σᵢuᵢvᵢᵀ stores a quarter of the numbers and the picture is already clear, because the singular values fall off fast.

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