Skip to main content
Calculator Answers

Eigenvalue Calculator

Blake Boege
Written by Blake Boege · Founder, Calculator Answers

An eigenvalue calculator is a linear algebra tool that computes the eigenvalues and eigenvectors of a square matrix. The calculator solves the characteristic equation, defined as the determinant of the matrix minus lambda times the identity matrix equals zero, to find the eigenvalues. It then calculates the corresponding eigenspaces and eigenvectors by solving the resulting null spaces. The calculator supports matrices up to size three-by-three and outputs step-by-step determinant expansion. College math and engineering students use this tool to study linear transformations.

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).

Quick Answer

Calculate eigenvalues and eigenvectors of a square matrix. Enter the matrix elements to find characteristic polynomials and unique vector solutions.

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

Eigenvalues (real, distinct)

λ₁ = 5, λ₂ = 2

λ² − (7)·λ + (10) = 0

Trace tr(A)7
Determinant det(A)10
Discriminant9
Characteristicλ² − (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.

Was this helpful?

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

Related Calculators

More tools from Education

Eigenvector CalculatorFind eigenvalues and unit eigenvectors of a 2×2 matrix. Returns trace, determinant, discriminant, both eigenvalues, and the corresponding unit eigenvectors.

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.