Overview
Once you have the eigenvalues, eigenvectors are the non-zero vectors that satisfy (A − λI)x = 0 for each eigenvalue λ. They represent the 'special directions' along which the matrix only scales, without rotating.
If eigenvalues tell you 'how much' the matrix stretches, eigenvectors tell you 'in which direction' it stretches. Together they completely characterize the matrix's behavior.
Eigenvectors
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Mathematical Foundation
Formal Statement
Required Conditions
- • First find the eigenvalues λ
- • For each λ, solve the homogeneous system (A − λI)x = 0
- • The solution space (null space) gives the eigenvectors
Common Mistakes & Pitfalls
- ⚠ Reporting the zero vector as an eigenvector — eigenvectors must be non-zero.
- ⚠ Not finding all linearly independent eigenvectors for a repeated eigenvalue.
- ⚠ Arithmetic errors in row reduction of (A − λI).