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Home/Math Visualization/PCA / SVD Geometry

PCA / SVD Geometry

Covariance ellipse and SVD view of dimensionality reduction: principal components, explained variance, and rank-1 reconstruction error.

Data cloud

2.2
0.75
32°
1

Measured values

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

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

Live graphs

About this model

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

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.