Linear Algebra

Singular Value Decomposition

Every matrix, however messy, is secretly just three clean steps: rotate, stretch along the axes, rotate again. That is the SVD, A = U Σ Vᵀ. Watch the unit circle turn into an ellipse and see exactly which rotation and which stretch produced it.

Matrix A

1.200.800.301.10

a 1.20

b 0.80

c 0.30

d 1.10

Stage of A = U Σ Vᵀ whole

Presets

Singular values σ₁, σ₂—

Condition σ₁/σ₂—

Determinant—

Rank—

What to observe

Drag the Stage slider left to right. The circle first rotates(Vᵀ), then stretches along the axes by σ₁ and σ₂ (Σ), thenrotates again (U). Three simple moves rebuild any matrix.
The teal and amber arrows are the right singular vectors v₁, v₂: the special input directions that stay perpendicular. A sends them to the ellipse's axes, scaled by the singular values σ₁ ≥ σ₂.
Push a preset toward Rank 1: σ₂ collapses to 0, the ellipse flattens to a line, and the determinant (σ₁σ₂) vanishes. The condition number σ₁/σ₂ blowing up is exactly what makes a matrix hard to invert.

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