Experiments / live

SVD playground

Every matrix does its work in three acts: rotate, stretch each axis on its own, rotate again. Set the four entries a, b, c, d of the matrix [[a, b], [c, d]] with the sliders below and hit play to watch the unit circle (faint) morph through the three acts — Vᵀ turning the marked axes, Σ pulling the circle open into an axis-aligned ellipse, U swinging it into place. The two heavy arrows are the singular directions; the readout gives the singular values σ₁, σ₂ — the lengths of the ellipse's semi-axes. Flip rank-1 to clamp σ₂ → 0 and watch the ellipse collapse to a line segment: the best rank-1 approximation of your matrix. The full story is in the essay.

the unit circle mapped to an ellipse, in three acts

a b c d

σ = … rank-1 (σ₂ → 0)

This playground is interactive and needs JavaScript. Here's the idea as a static picture: a 2×2 matrix maps the unit circle to an ellipse in three acts — Vᵀ rotates (the marked axes turn), Σ stretches the circle into an axis-aligned ellipse with semi-axes σ₁ and σ₂, and U rotates the ellipse into its final orientation.