Eigenvalues & Eigenvectors Calculator
Find the eigenvalues and eigenvectors of a square matrix. Analytical computation for 2×2 and 3×3 matrices. Complex eigenvalues are shown in a+bi form.
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Notes
What are Eigenvalues and Eigenvectors?
A scalar λ is an eigenvalue of A, and nonzero vector v is the corresponding eigenvector, if Av = λv. Eigenvalues are roots of det(A − λI) = 0.
Worked 2×2 Example
Find eigenvalues of A = [[4,1],[2,3]]. Form A − λI:
| 4-λ | 1 |
| 2 | 3-λ |
Characteristic polynomial: (4−λ)(3−λ) − 2 = λ² − 7λ + 10 = 0
Verify: λ₁ + λ₂ = 7 = tr(A) ✓. λ₁ · λ₂ = 10 = det(A) ✓.
Real symmetric matrices always have real eigenvalues. Non-symmetric matrices may have complex eigenvalues as conjugate pairs.
See also
- Matrix Calculations Guide — Eigenvalues & Eigenvectors — In-depth notes on Eigenvalues & Eigenvectors with worked examples
Frequently Asked Questions
What is the characteristic polynomial?
det(A − λI) = 0 is the characteristic equation. Its roots are the eigenvalues of A.
Can eigenvalues be complex?
Yes. Non-symmetric real matrices can have complex eigenvalues that come in conjugate pairs (a ± bi).
What is the relationship between eigenvalues and the trace and determinant?
For an n×n matrix: trace(A) = sum of eigenvalues, det(A) = product of eigenvalues.