Eigenvalues & eigenvectors, visualized
Most vectors get knocked off course when a matrix acts on them — rotated and stretched. But a few special directions, the eigenvectors, only get stretched, never turned. The amount each one is stretched is its eigenvalue. Together they are the natural axes of a transformation.
This one idea powers Principal Component Analysis, PageRank, the stability of dynamical systems, and the curvature of loss surfaces. Drag the matrix entries and the dashed arrows show the eigenvectors — the directions the matrix leaves pointing the same way.
Drag the matrix entries
Determinant (area scale)
det = 3
area grows by this factor
Eigenvalues
3, 1
dashed arrows = directions only stretched, never rotated
Free · runs entirely in your browser · nothing to install
How to use it
- Drag the four matrix entries and watch the blue unit square get stretched, sheared, or flipped.
- Follow the dashed amber arrows — those are the eigenvectors, the directions that never rotate.
- Read the eigenvalues: each one is how much its eigenvector gets stretched (negative means flipped).
What you'll take away
- What makes a direction an 'eigenvector' — it's only scaled, not rotated.
- How the eigenvalue is the stretch factor along that direction.
- Why symmetric matrices give perpendicular eigenvectors — the fact PCA relies on.
Want to actually build this?
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FAQ
- What is an eigenvector in simple terms?
- It's a direction that a matrix only stretches or shrinks, without rotating it. Multiplying the matrix by that vector gives back the same vector scaled by a number — the eigenvalue.
- What are eigenvalues used for in machine learning?
- They are central to PCA, where the eigenvectors of the data's covariance matrix are the principal directions and the eigenvalues are the variance captured along each — letting you compress data while keeping the most information.