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Lab · c09.s06.L1

SVD rotate-stretch

Rotate, stretch, rotate: only the stretch in the middle changes any length.

Question: What does every matrix do, in three steps?

SVD rotate-stretchA unit circle on a grid passes through three moves. First it turns rigidly, radius unchanged. Then it stretches along two perpendicular directions into an ellipse whose half-axes are the two singular values. Finally it turns rigidly again, tilting that ellipse. The outline never bends or dents: only the middle step changes any distance, and a tiny second singular value flattens the ellipse toward a segment.SVD rotate-stretchA = U Σ Vᵀ · only the middle step changes lengthsxyStartRotate inStretchRotate outShear[[1.00, 1.00], [0.00, 1.00]]σ₁ = 1.62σ₂ = 0.62stretch ratio = 2.62rot in = -58.28°rot out = 31.72°net rot = -26.57°stage 0.00 of 3.00 · startStart
Re-simulated Re-simulated: four chosen 2×2 matrices are decomposed exactly by data/scripts/c09.s06.L1.py. The matrices are chosen to be read, not measured. A = U Σ Vᵀ holds to machine precision. All four have positive determinant, so both outer factors are genuine rotations; in general one of them may be a reflection. The statement holds in every dimension, and the section's 4×3 worked example shows a matrix that is not square.
Lab controls
Shear

Arrow keys change preset; Home and End jump to the first or last

0 stage

Left/Right 0.25, Up/Down 0.05, Page Up/Down 1.00, Home 0.00, End 3.00

On

Space toggles when focused; key G toggles anywhere

Frame 1 of 13: Start: the unit circleThe unit circle sits on the grid with radius 1, so every point on it is an input. The matrix is the Shear preset, [[1.00, 1.00], [0.00, 1.00]]. The gold spoke marks one point so that a turn will be visible.

Keyboard: Space plays or pauses, Left and Right step, Home goes to the first frame, End goes to the last frame.

Text description of this lab

A unit circle on a grid passes through three moves. First it turns rigidly, radius unchanged. Then it stretches along two perpendicular directions into an ellipse whose half-axes are the two singular values. Finally it turns rigidly again, tilting that ellipse. The outline never bends or dents: only the middle step changes any distance, and a tiny second singular value flattens the ellipse toward a segment.

Return to the section that hosts this lab

Caption status: re-simulated. Every lab declares one of published, re-simulated, schematic, synthetic or interpolated.