How do I find eigenvalues of a 2×2 matrix?

Solve λ² − (a+d)λ + (ad−bc) = 0 for A = [[a,b],[c,d]]. Then solve (A − λI)v = 0 for each root if eigenvectors are needed.

Does this calculator also explain eigenvectors?

Yes. Ask for an eigenspace basis and a verification of Av = λv. Different nonzero scalar multiples can describe the same eigenvector direction.

What if an eigenvalue is repeated?

Its algebraic multiplicity counts repeated polynomial roots. Its geometric multiplicity is the eigenspace dimension; these need not be equal.

Can a real matrix have complex eigenvalues?

Yes. For example, [[0,−1],[1,0]] has eigenvalues i and −i. State whether the solution should be over the real or complex numbers.

Can I enter a 3×3 matrix and get exact results?

You can ask for a characteristic polynomial and worked explanation of a 3×3 or larger square matrix. Results are AI-generated, so exact symbolic or numerical accuracy is not guaranteed. Verify the polynomial and eigenpairs independently.

Matrices, methods, and checks

Eigenvalue Calculator

Use this eigenvalue calculator for AI explanations of characteristic polynomials, eigenvectors, repeated roots, and complex eigenvalues. Paste a square matrix and check Av = λv.

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Enter a problem

eigenvalue calculator

Av = λv

Characteristic polynomial

Eigenvectors and eigenspaces

Repeated eigenvalues

eigenvalue calculator

Work through your eigenvalue problem

Paste each matrix row in brackets, or upload a clear image. State the operation and ask for steps. Responses are AI-generated; check important arithmetic independently.

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eigenvalue calculator

From a square matrix to eigenvalues and eigenspaces

Find the roots, solve for the corresponding vectors, and keep multiplicity and the number system in view.

01p(λ) = det(λI − A)

Characteristic polynomial

Form det(λI − A), then set it equal to zero. For a 2×2 matrix, use trace and determinant to check the quadratic.

02(A − λI)v = 0

Eigenvectors and eigenspaces

For each eigenvalue, solve (A − λI)v = 0. Eigenvectors must be nonzero; any nonzero scalar multiple is also valid.

031 ≤ dim Eλ ≤ mλ

Repeated eigenvalues

Compare algebraic multiplicity with eigenspace dimension. A repeated eigenvalue alone does not decide diagonalizability.

04A = PDP⁻¹

Complex roots and diagonalization

State whether you work over ℝ or ℂ. A real matrix can have complex eigenvalues; diagonalization needs a full eigenvector basis.

Method

How to use the eigenvalue calculator

Keep the original matrix visible while you follow the explanation.

1
Enter the square matrix
A ∈ ℝⁿˣⁿ
Keep every row and sign. Specify real or complex scalars and whether you need eigenvectors.
2
Form the polynomial
det(A − λI) = 0
Subtract λ on the diagonal and evaluate det(A − λI). Its zeros give the eigenvalues.
3
Solve each eigenspace
v ∈ ker(A − λI), v ≠ 0
For every root, row-reduce A − λI and choose nonzero vectors from its null space.
4
Verify the eigenpairs
Av − λv = 0
Check Av = λv. Count independent eigenvectors if diagonalization is requested.

Tip: det(A − λI) and det(λI − A) differ by (−1)ⁿ, but have the same roots. Keep one polynomial convention throughout.

Quick reference

Eigenvalue formulas and checks

Count eigenvalues with algebraic multiplicity when checking their sum and product.

Eigenpair

Av = λv, v ≠ 0

An eigenvalue scales a corresponding nonzero eigenvector.

2×2 polynomial

p(λ) = λ² − tr(A)λ + det(A)

For A = [[a,b],[c,d]], tr(A) = a + d and det(A) = ad − bc.

Sum and product

Σλᵢ = tr(A); ∏λᵢ = det(A)

Include repeated roots in both checks, and complex roots when needed.

Diagonalization

AP = PD

Columns of P must be n linearly independent eigenvectors; D holds the matching eigenvalues.

Worked example

Worked eigenvalue examples

These reviewed examples show the method, result, and an independent check. Use the example prompts above to try them in the solver.

Worked example 1

Eigenpairs of A = [[3,1],[0,2]]

A triangular matrix with two distinct eigenvalues.

p(λ) = (λ−3)(λ−2), so λ = 3 and λ = 2.
For λ = 3, (A−3I)v = 0 gives y = 0; choose v = [1,0].
For λ = 2, (A−2I)v = 0 gives x + y = 0; choose v = [−1,1].

Result: λ = 3: span{[1,0]}; λ = 2: span{[−1,1]}

Check: A[1,0] = [3,0]; A[−1,1] = [−2,2] = 2[−1,1]. Sum 5 = tr(A); product 6 = det(A).

Worked example 2

Repeated root: A = [[2,1],[0,2]]

Two equal roots do not guarantee two independent eigenvectors.

p(λ) = (λ−2)², so λ = 2 has algebraic multiplicity 2.
A−2I = [[0,1],[0,0]], so y = 0 and the eigenspace is span{[1,0]}.
Its eigenspace has dimension 1, so there are not two independent eigenvectors.

Result: λ = 2 (multiplicity 2); A is not diagonalizable.

Check: A[1,0] = 2[1,0]. The trace is 4 and the determinant is 4, consistent with roots 2, 2.

Worked example 3

Complex roots: A = [[0,−1],[1,0]]

A quarter-turn rotation has no real eigenvalues.

p(λ) = λ² + 1, so the complex roots are i and −i.
For λ = i, choose v = [1,−i]; for λ = −i, choose v = [1,i].
Over ℝ there are no eigenvectors; over ℂ these two vectors are independent.

Result: λ = i, −i; diagonalizable over ℂ, not over ℝ.

Check: A[1,−i] = [i,1] = i[1,−i]; the sum is 0 and product is 1.

Common mistakes to catch

Check the conditions before trusting a result.

Using the zero vector

The zero vector satisfies Av = λv for every λ, so it is excluded from the definition of eigenvector.

Equating multiplicity with eigenspace dimension

A repeated root may have too few independent eigenvectors. Solve its null space before deciding diagonalizability.

Discarding complex roots

A real matrix need not have real eigenvalues. Specify the field and check all roots with multiplicity.