Every real matrix A can be decomposed as:
A = U Σ Vᵀ
U (m×m) — left singular vectors (orthogonal
columns, rotation/reflection) · Σ (m×n) —
diagonal matrix of singular values σ₁ ≥ σ₂ ≥ 0 ·
Vᵀ (n×n) — right singular vectors (orthogonal
rows, rotation/reflection).
Geometric interpretation: A maps the unit circle
to an ellipse with semi-axes σ₁ (along u₁) and σ₂
(along u₂). The columns of V are
the input directions that get the largest scaling. For det > 0
the transformation preserves orientation; det < 0 flips it.
Applications: principal component analysis (PCA),
image compression, pseudo-inverse, noise reduction, latent
semantic analysis. The nuclear norm
‖A‖* = Σ σᵢ is the trace-norm used in low-rank regularization.
🔢 SVD Visualizer
About this simulation
Singular Value Decomposition factors any real matrix into a rotation, a
pure stretch and another rotation — the single most useful matrix
factorisation in applied mathematics. This visualizer shows the geometry
directly: edit a 2×2 matrix and watch the unit circle on the left map to
an ellipse on the right. The same decomposition powers principal-component
analysis, image compression, recommendation engines and the noise
reduction behind countless data-science pipelines.
How it works
The right singular vectors v₁, v₂ are drawn as the input directions that get stretched the most.
Matrix A maps the dashed unit circle into an ellipse whose semi-axes are the singular values σ₁ and σ₂.
The ellipse axes point along the left singular vectors u₁, u₂, scaled by their singular values.
The bar chart shows each σ and how much of the matrix's energy the largest singular value captures.
Key equations
A = U·Σ·Vᵀ — U holds the left singular vectors (output rotation),
Σ is diagonal with σ₁ ≥ σ₂ ≥ 0, and Vᵀ holds the right singular vectors (input rotation).
The condition number cond = σ₁ / σ₂ and the spectral norm ‖A‖₂ = σ₁.
Controls
Presets — load identity, scaling, shear, rotation, low-rank or random matrices.
a, b, c, d inputs — type each entry of the 2×2 matrix A and see the SVD update live.
Stats bar — read σ₁, σ₂, determinant, rank, spectral norm and condition number.
? — open this information panel.
Did you know?
A singular value of zero means the matrix collapses a whole direction to
nothing — that is exactly when its rank drops and the transformation can
never be reversed, which is why SVD is the reliable way to detect and
handle near-singular matrices in scientific computing.