How Eigenvalues Work
Linear Transformations
A matrix stretches, shears, and rotates space
The determinant is greater than 1 — this matrix expands area. Transformed points spread outward from the origin.
Step 1 of 6: Linear Transformations
A matrix stretches, shears, and rotates space
The determinant is greater than 1 — this matrix expands area. Transformed points spread outward from the origin.
Step 2 of 6: Special Directions
Eigenvectors only scale — they never rotate
Eigenvectors are special directions that only scale under the matrix — they never rotate. The pink (v₁) and purple (v₂) arrows stay on the same line after transformation. All other vectors (gray) change direction. Adjust the matrix to see how eigenvectors shift.
Step 3 of 6: Eigenvalues as Scale Factors
How much each eigenvector stretches or flips
Both eigenvalues are positive — the matrix stretches space along both eigenvector directions without flipping. This is a pure stretch/compression.
Step 4 of 6: The Characteristic Equation
Finding eigenvalues by solving det(A - λI) = 0
The discriminant is moderate — two distinct real eigenvalues. The characteristic polynomial crosses zero at two separate points.
Step 5 of 6: Eigendecomposition
A = PDP⁻¹ — rotate, scale, rotate back
Phase 1: P⁻¹ rotates the space so eigenvectors align with the axes. This reveals the 'natural coordinate system' of the matrix.
Step 6 of 6: Why Eigenvectors Matter
Principal Component Analysis finds the directions of most variance
Clear dominant direction! PC1 captures most of the variance. You could project onto PC1 alone and keep most information — that's the power of PCA.