Eigenvalue Calculator
Calculate the eigenvalues of a 2×2 matrix using the characteristic equation.
Inputs
Eigenvalue λ₁
5.000000
Eigenvalue λ₂
2.000000
Trace
7.000000
Determinant
10.000000
Step by step
Trace (a + d)
4 + 3
= 7.000000
Determinant (ad − bc)
4×3 − 1×2
= 10.000000
Discriminant: trace² − 4×det
7.00² − 4×10.00
= 9.000000
λ₁ = (trace + √discriminant) / 2
(7.00 + √9.00) / 2
= 5.000000
λ₂ = (trace − √discriminant) / 2
(7.00 − √9.00) / 2
= 2.000000
How it works
Eigenvalues of a 2×2 matrix are found by solving the characteristic equation det(A − λI) = 0, which reduces to λ² − trace×λ + det = 0. Using the quadratic formula: λ = (trace ± √(trace² − 4×det)) / 2. Eigenvalues are fundamental in PCA, spectral clustering, and stability analysis of dynamical systems.
Formula
Characteristic Equation
lambda = (trace ± sqrt(trace^2 - 4*det)) / 2
- trace
- Sum of diagonal elements (a + d)
- det
- Determinant of the matrix
Frequently Asked Questions
How are eigenvalues used in PCA?
In PCA, eigenvalues of the covariance matrix represent the variance explained by each principal component. The largest eigenvalues correspond to the directions of maximum variance in the data.
What if the discriminant is negative?
A negative discriminant means the eigenvalues are complex conjugates. This occurs in matrices representing rotations or oscillatory systems and indicates no real principal directions exist.