Overview
Eigenvalues are special scalars λ associated with a square matrix A. They represent the factors by which the matrix stretches or compresses space along certain directions (eigenvectors). You find them by solving det(A − λI) = 0.
When a matrix transforms a vector and the result is just a scaled version of the original vector (same direction, just longer or shorter), the scaling factor is the eigenvalue. The direction that doesn't change is the eigenvector.
Eigenvalues
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Mathematical Foundation
Formal Statement
Required Conditions
- • A must be a square matrix
- • The characteristic polynomial is of degree n for an n×n matrix
- • Eigenvalues can be real or complex numbers
Common Mistakes & Pitfalls
- ⚠ Sign errors when computing (A − λI) — be careful with the diagonal subtraction.
- ⚠ Arithmetic mistakes in expanding the determinant, especially for 3×3 matrices.
- ⚠ Forgetting that repeated eigenvalues are possible (algebraic multiplicity).