singular-value-decomposition #

Shows the geometric meaning of SVD by animating the unit circle in the domain as it is transformed step-by-step by Vᵀ (rotation), then Σ (nonnegative axis scaling by singular values), then U (rotation) into an ellipse. The left panel displays right singular vectors (columns of V) on the unit circle; the right panel displays the resulting ellipse with its principal axes aligned to left singular vectors (columns of U) and lengths proportional to σ₁ and σ₂. A step strip reinforces A = UΣVᵀ and the relationship σᵢ = √λᵢ for eigenvalues of AᵀA.

canvasclick to interact

⏮◀◀▶▶STEP0.25x1xZOOM

t=0s

practical uses #

01.Low-rank approximation and compression (PCA-style dimensionality reduction)
02.Solving least squares problems and computing pseudoinverses
03.Stability/conditioning analysis via singular values and spectral norm

technical notes #

Pure Canvas2D, responsive scaling via scale=min(w,h)/240 and grid snapping for a blocky look. Uses a guaranteed 2×2 example matrix built from chosen U, Σ, V to ensure an exact SVD. The animation interpolates between partial transforms (I→Vᵀ→ΣVᵀ→UΣVᵀ) over a 4.2s cycle using the provided cubic ease(t).

← logistic-regression convex-optimization →