Education
Eigenvalue Calculator
Enter the four entries of a 2×2 matrix. The calculator returns the trace, determinant, characteristic polynomial λ² − tr(A)·λ + det(A) = 0, and the two eigenvalues (real, repeated, or complex).
2 × 2 matrix A
Enter the four entries of a 2×2 matrix. The calculator returns the eigenvalues using the trace and determinant identity.
Scope
This calculator handles 2×2 matrices. The characteristic equation simplifies to a quadratic in λ:
λ² − tr(A)·λ + det(A) = 0
For 3×3 and larger matrices, eigenvalue computation needs a cubic (or higher) characteristic polynomial solver. That is out of scope on this page; use a CAS or numerical library.
Eigenvalues (real, distinct)
λ₁ = 5, λ₂ = 2
λ² − (7)·λ + (10) = 0
Real distinct eigenvalues correspond to a matrix with two independent directions of pure scaling. Complex eigenvalues indicate a rotation component. A repeated eigenvalue means a single eigenvalue with algebraic multiplicity 2.
Examples
A = [[4,1],[2,3]]
λ = 5 and 2
A = [[2,1],[0,2]]
λ = 2 (repeated)
A = [[0,−1],[1,0]] (rotation)
λ = ±i
A = [[3,2],[1,4]]
λ ≈ 5 and 2
How it works
For any 2×2 matrix A, the characteristic equation det(A − λI) = 0 expands to a quadratic whose coefficients are the trace and determinant. Solving with the quadratic formula gives the eigenvalues directly.
Characteristic · λ² − tr(A)·λ + det(A) = 0
Discriminant · D = tr(A)² − 4·det(A)
Eigenvalues · λ = (tr(A) ± √D) / 2
Related linear algebra calculators
- Matrix calculator for matrix arithmetic on the matrix you are analyzing.
- Quadratic formula calculator for the underlying equation when you want to see the full quadratic worked.
- RREF calculator for finding eigenvectors once you have the eigenvalues (solve (A − λI)x = 0).
- All education calculators.
Frequently asked questions
An eigenvalue λ of a matrix A is a scalar such that Ax = λx for some nonzero vector x. The vector x is called an eigenvector. Eigenvalues capture the matrix's scaling behavior along its principal directions.
Solve the characteristic equation det(A − λI) = 0, which simplifies to λ² − tr(A)·λ + det(A) = 0 for any 2×2 matrix. Use the quadratic formula on this single equation.
The 2×2 case reduces to a quadratic with a clean closed-form solution. 3×3 needs a cubic characteristic polynomial, and larger sizes need iterative numerical methods (QR algorithm, etc.). For higher sizes, use a computer algebra system or a numerical library.
Complex eigenvalues come in conjugate pairs and indicate that the matrix represents a rotation (or rotation-plus-scaling). The calculator reports them as a ± bi with the real and imaginary parts.
When the discriminant is exactly zero, both roots of the characteristic equation collapse to the same value. This eigenvalue has algebraic multiplicity 2. Geometric multiplicity (the number of independent eigenvectors) can be 1 or 2 depending on the matrix.
Yes. tr(A) = A₁₁ + A₂₂. It equals the sum of the eigenvalues (counted with multiplicity). The determinant equals their product, by Vieta's formulas on the quadratic.
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