PhysSandbox
Classical MechanicsWaves & SoundElectricity & MagnetismOptics & LightGravity & OrbitsLabs
🌙Astronomy & The Sky🌡️Thermodynamics🌍Biophysics, Fluids & Geoscience📐Math Visualization🔧Engineering🧪Chemistry

More from Math Visualization

Other simulators in this category — or see all 84.

View category →
NewUniversity / research

Gradient Descent Optimizers

Compare SGD, momentum, and Adam on a curved loss landscape; tune learning rate, curvature, stability, and iteration count.

Launch Simulator
NewUniversity / research

Gaussian Mixture EM Algorithm

Expectation-maximization for a two-component Gaussian mixture: responsibilities, E/M steps, likelihood, and covariance ellipses.

Launch Simulator
NewUniversity / research

Support Vector Machine Margin

Hard/soft-margin SVM sketch with C penalty, hinge loss, support vectors, decision boundary, and margin bands.

Launch Simulator
NewUniversity / research

Bayesian Updating / Conjugate Priors

Beta-binomial Bayesian updating with prior/posterior curves, posterior predictive probability, and credible intervals.

Launch Simulator
NewUniversity / research

Markov Chain Mixing

Three-state Markov chain with transition matrix, stationary distribution, total variation distance, detailed-balance cue, and spectral-gap proxy.

Launch Simulator
NewUniversity / research

LMS / NLMS Adaptive Noise Cancellation

Primary p = s + v with v a fixed unknown FIR of Gaussian reference x[n]. Watch an L-tap FIR adapt by LMS or NLMS so error e = p − wᵀx → s; running MSE and ‖w − h‖.

Launch Simulator
PhysSandbox

Interactive physics, chemistry, and engineering simulators for students, teachers, and curious minds.

Physics

  • Classical Mechanics
  • Waves & Sound
  • Electricity & Magnetism

Science

  • Optics & Light
  • Gravity & Orbits
  • Astronomy & The Sky

More

  • Thermodynamics
  • Biophysics, Fluids & Geoscience
  • Math Visualization
  • Engineering
  • Chemistry

© 2026 PhysSandbox. Free interactive science simulators.

PrivacyTermsContact
Home/Math Visualization/PCA / SVD Geometry

PCA / SVD Geometry

PCA rotates a data cloud into orthogonal directions of maximum variance. The simulator draws a covariance ellipse, principal component axes, explained-variance ratios, and the rank-1 reconstruction obtained by projecting every point onto PC1. In SVD language, the singular values determine how much variance each right singular vector explains.

Who it's for: Linear algebra, statistics, data science, machine learning, signal processing, and numerical methods courses.

Key terms

  • PCA
  • SVD
  • Covariance matrix
  • Principal component
  • Explained variance
  • Low-rank reconstruction

Yellow points show rank-1 reconstruction: data projected onto the first principal component.

Live graphs

Data cloud

2.2
0.75
32°
1

Measured values

PC1 variance89.6%
PC2 variance10.4%
Reconstruction RMSE0.750

How it works

PCA and SVD geometry: covariance ellipse, principal components, explained variance, and rank-1 reconstruction.

Key equations

X = UΣVᵀ; covariance eigenvectors give principal axes
Explained variance ratio = σ_i² / Σ σ_j²

Frequently asked questions

Why are the principal components perpendicular?
For a covariance matrix, eigenvectors belonging to distinct eigenvalues are orthogonal because the matrix is symmetric. PCA uses those eigenvectors as the rotated coordinate axes.
What does reconstruction error mean?
If only the first component is kept, all variation along the discarded component is lost. The remaining spread along PC2 is a geometric picture of rank-1 reconstruction error.