λEigenvalue Calculator (2×2 Matrix)
Enter matrix [[a, b], [c, d]]
Eigenvalues and eigenvectors are fundamental concepts in linear algebra. For a square matrix A, an eigenvector v is a non-zero vector that only scales (not rotates) when A is applied: Av = λv. The scalar λ is the corresponding eigenvalue. Eigenvalues reveal the fundamental behaviour of linear transformations and appear throughout physics, engineering, statistics (PCA), and machine learning.
- 1For 2×2 matrix [[a,b],[c,d]], eigenvalues satisfy the characteristic equation: det(A − λI) = 0
- 2det(A − λI) = λ² − (a+d)λ + (ad−bc) = 0
- 3Trace = a+d, Determinant = ad−bc
- 4λ = [Trace ± √(Trace² − 4·Det)] / 2
- 5Discriminant > 0: two real eigenvalues; < 0: complex conjugate pair
Matrix [[4,1],[2,3]]=λ₁=5, λ₂=2Trace=7, Det=10, Disc=9
Matrix [[0,−1],[1,0]] (rotation 90°)=λ = ±i (complex)Pure rotation has no real eigenvalues
| Eigenvalue Type | Geometrical Meaning |
|---|---|
| λ > 1 | Stretches in eigenvector direction |
| 0 < λ < 1 | Compresses in eigenvector direction |
| λ = 1 | No change along eigenvector |
| λ = 0 | Matrix is singular (det = 0); projects onto a subspace |
| λ < 0 | Reflects and possibly scales |
| Complex λ | Rotation component; no real invariant direction |
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