Eigenvalue Calculator
Find the eigenvalues of a 2×2 matrix from its characteristic equation.
Characteristic equation
λ² − 0·λ + 0 = 0
Discriminant Δ = (trace)² − 4·(det) = 0. Because Δ ≥ 0 both eigenvalues are real.
How it works
- 1Enter the four entries of your 2×2 matrix A into the grid.
- 2The calculator solves λ² − tr(A)·λ + det(A) = 0 using the quadratic formula to find λ₁ and λ₂.
- 3If the discriminant is negative, the result is a complex-conjugate pair displayed as a ± bi.
Use cases
- Checking stability of a 2D linear dynamical system — negative real parts mean the origin is stable.
- Principal component analysis: eigenvalues of a 2×2 covariance matrix reveal variance along each axis.
- Verifying hand-computed characteristic equations during linear-algebra coursework.
Frequently asked questions
What is an eigenvalue?
A scalar λ such that Av = λv for some non-zero vector v. Geometrically, λ describes how the matrix A stretches or shrinks space along the eigenvector direction.
When are the eigenvalues complex?
When the discriminant (trace)² − 4·(det) is negative. The two values form a conjugate pair a + bi and a − bi, which represents a rotation-scaling transformation.
Can I use this for larger matrices?
This tool is fixed to 2×2 matrices because the quadratic formula gives exact closed-form roots. For 3×3 and larger, numerical methods are required.
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