SVD Geometry
Decompose a 2×2 matrix into rotations/reflection and singular-axis scaling.
About this tool
Enter a 2×2 matrix or select a preset. The picture and singular values update immediately. Blue is the unit circle, gold its image under A. Dashed red/green directions are v₁,v₂; solid matching directions are σ₁u₁,σ₂u₂. The image has semiaxes σ₁≥σ₂≥0. A rank-one map collapses the circle to a line segment; a zero map to a point. PNG saves the current picture.
A=UΣVᵀ: V chooses orthogonal input directions, Σ stretches them, U supplies orthogonal output directions including reflections. Decomposition opens U,V,reconstruction, determinant and a normwise relative error. Repeated singular values and zero directions do not have unique vectors. Numerical rank uses σ₂/σ₁>1e−12, while a nonzero matrix always has at least rank one.
The plot uses common coordinate units and fits the largest singular radius and the unit circle without distortion. Very small features can be subpixel; inspect the numeric values. Finite decimal or scientific inputs are accepted without clipping. Scaled calculation avoids squared-norm overflow and subtraction of nearly equal eigenvalues. Float64 and ten display significant digits; extreme relative scales/cancellation can still lose precision. Unrepresentable determinants are marked rather than confused with zero. Valid restored inputs are calculated again; only raw inputs and the chosen preset are stored. Use Decomposition for the factors and numerical diagnostics; colour meanings are explained above.