Eigenvalues & Eigenvectors
Eigenvalues λ and eigenvectors v satisfy Av = λv. Eigenvectors are directions unchanged by the transformation; eigenvalues are the scaling factors. Solve det(A − λI) = 0 for eigenvalues.
Why This Mathematical Concept Matters
Why: Eigenvalues power PCA, PageRank, quantum mechanics, and vibration analysis. They reveal a matrix's fundamental behavior.
How: Compute characteristic polynomial det(A − λI) = 0. For 2×2: λ² − tr(A)λ + det(A) = 0. For each λ, solve (A − λI)v = 0 for v.
- ●Symmetric matrices: all eigenvalues real.
- ●Diagonal matrices: eigenvalues = diagonal entries.
- ●Zero eigenvalue ⟺ singular matrix.
Quick Examples — Click to Load
Matrix A (2×2)
λ² - 4λ + 3 = 0
Eigenvalues (Bar)
Eigenvalue Magnitudes (Doughnut)
Calculation Steps
⚠️For educational and informational purposes only. Verify with a qualified professional.
🧮 Fascinating Math Facts
PCA uses eigenvectors of covariance
Rotation matrices: complex eigens
Key Takeaways
- • Av = λv — eigenvector v scaled by eigenvalue λ.
- • Characteristic polynomial: det(A - λI) = 0.
- • 2×2: λ² - tr(A)λ + det(A) = 0.
- • Trace = sum of eigenvalues; Determinant = product.
- • Symmetric matrices have real eigenvalues.
Did You Know?
How It Works
1. Characteristic polynomial: Solve det(A - λI) = 0. For 2×2: λ² - tr(A)λ + det(A) = 0.
2. Eigenvalues: Roots of the characteristic polynomial.
3. Eigenvectors: For each λ, solve (A - λI)v = 0.
2×2: λ² - (a+d)λ + (ad-bc) = 0
tr(A) = a+d, det(A) = ad-bc
Expert Tips
Diagonal
Diagonal matrix: eigenvalues = diagonal entries.
Triangular
Upper/lower triangular: eigenvalues = diagonal.
Symmetric
All eigenvalues real; eigenvectors orthogonal.
Zero Eigenvalue
A singular ⟺ det(A)=0 ⟺ 0 is an eigenvalue.
Comparison Table
| Feature | This Calculator | NumPy | Manual |
|---|---|---|---|
| 2×2 & 3×3 support | ✅ | ✅ | ⚠️ |
| Characteristic polynomial | ✅ | ❌ | ⚠️ |
| Bar & Doughnut charts | ✅ | ❌ | ❌ |
| 8 preset examples | ✅ | ❌ | ❌ |
FAQ
What is an eigenvalue?
Scalar λ such that Av = λv for some non-zero v.
What is an eigenvector?
Non-zero v such that Av = λv (direction unchanged).
What is the characteristic polynomial?
det(A - λI) = 0. Roots are eigenvalues.
How do trace and determinant relate?
tr(A) = sum of eigenvalues; det(A) = product.
Can eigenvalues be complex?
Yes. Real matrices: complex eigenvalues come in pairs.
What is a repeated eigenvalue?
Algebraic multiplicity > 1. May have fewer eigenvectors.
When is a matrix diagonalizable?
When it has n linearly independent eigenvectors.
What about symmetric matrices?
All eigenvalues real; eigenvectors orthogonal.
Stats
Sources
- • Gilbert Strang, Linear Algebra and Its Applications
- • Khan Academy: khanacademy.org
- • MIT 18.06: ocw.mit.edu
- • Wolfram MathWorld: mathworld.wolfram.com
- • 3Blue1Brown: 3blue1brown.com
- • Numerical Recipes: Eigenvalue algorithms
Disclaimer: For educational purposes. Uses JavaScript floating-point. Complex eigenvalues shown as real part only. Verify critical calculations independently.