Linear Algebra · Rotate, stretch, rotate

SVD as stretch

A 2×2 SVD: the unit circle becomes an ellipse. σ₁ and σ₂ are lengths, not magic.

θ — input rotation, V (Vᵀ undoes it)0.40σ₁ — stretch along v₁ (solid)2.00σ₂ — stretch along v₂ (dashed)0.70φ — output rotation, U0.20

A = UΣVᵀ: 1.860.640.100.79
det A = σ₁σ₂: 1.40
rank: 2

Thin green lines on the circle are v₁ and v₂, the input directions. Vᵀ turns them onto the axes, Σ stretches by σ₁ and σ₂, U turns the result by φ. The light axes of the ellipse are σ₁u₁ and σ₂u₂. Rotations never change lengths, so the ellipse's half-axes are exactly σ₁ and σ₂.

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SVD as stretch

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