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Interactive 3D/Eigenvalues and Eigenvectors
→ Input·→ Transformed·→ Eigenvectors
2×2 Matrix
Presets
Symmetric matrices always have real eigenvectors. PCA uses them.
Actions
Eigenvalues
λ₁ = 3.6180
λ₂ = 1.3820
v₁ = [-0.851, -0.526]
v₂ = [-0.526, 0.851]
About
Eigenvectors are the "stable directions" of a transformation. PCA finds the eigenvectors of the covariance matrix - they become your principal components.
Try: Switch to "Rotation" and notice there are no green eigenvectors - the matrix has no real stable direction.

Eigenvalues and Eigenvectors - Interactive Visualization

Eigenvalues and eigenvectors are the most important concept in applied linear algebra. When you apply a matrix transformation, most vectors change direction - but eigenvectors only scale, they never rotate. This interactive demo shows exactly which directions are stable under any 2×2 matrix you choose.

  • PCA finds the eigenvectors of the covariance matrix - they become your principal components
  • The eigenvalue λ tells you how much each eigenvector stretches or compresses
  • Complex eigenvalues mean no real eigenvectors - pure rotation with no stable directions
  • Symmetric matrices always have real, orthogonal eigenvectors (why covariance matrices are well-behaved)
  • Used in: PCA, spectral clustering, PageRank, stability analysis of neural networks
  • Try the "Rotation" preset to see why rotating matrices have no real eigenvectors

Part of the EngineersOfAI Interactive 3D - free interactive visualizations covering every major concept in machine learning and AI engineering. Hover any element for a plain-English explanation. No code required.