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Matrix Eigenvectors Calculator

Find the eigenvectors of a matrix in seconds. Calculate real eigenvalues, determine eigenspaces, handle repeated eigenvalues, and follow the solution step by step.

Step-by-step solution 1×1 and 2×2 matrices No sign-up required
THE EIGENVECTOR EQUATION
Av = λv

A matrix transforms an eigenvector by scaling it.

This calculator finds real eigenvectors for square matrices of size 1×1 and 2×2. Complex eigenvalues are identified when no real eigenvectors exist.

Calculate Eigenvectors

Enter the entries of a square matrix.

Real-valued input

Enter finite numbers between −1,000,000 and 1,000,000.

What you will get

  • Real eigenvalues and their associated eigenvectors.
  • A convenient representative for each eigenvector direction.
  • The dimension of each eigenspace when it can be determined.
  • The trace, determinant, and characteristic-equation discriminant.
  • A step-by-step explanation of the calculation.
METHOD
1. Find the eigenvalues:
det(A − λI) = 0
2. Find the eigenvectors:
(A − λI)v = 0

An eigenvector must be nonzero. Multiplying an eigenvector by any nonzero scalar gives another eigenvector for the same eigenvalue.

Formula for Finding Eigenvectors

An eigenvector is a nonzero vector that keeps its direction under a linear transformation, although its magnitude can change.

Av = λv

A is a square matrix, v is a nonzero eigenvector, and λ is its eigenvalue.

Step 1: Find the eigenvalues

Rearrange the eigenvector equation to obtain (A − λI)v = 0. A nonzero solution exists only when the matrix A − λI is singular. Therefore, solve the characteristic equation det(A − λI) = 0.

Step 2: Solve for each eigenvector

Substitute each eigenvalue into A − λI, then solve the homogeneous linear system. The solutions form the eigenspace associated with that eigenvalue. Every nonzero vector in this space is an eigenvector.

For a 2×2 matrix

Let

A = [[a, b], [c, d]]

The characteristic equation is

λ² − (a + d)λ + (ad − bc) = 0

The discriminant is

Δ = (a + d)² − 4(ad − bc)
  • Δ > 0: two distinct real eigenvalues.
  • Δ = 0: a repeated real eigenvalue; its eigenspace may be one- or two-dimensional.
  • Δ < 0: complex-conjugate eigenvalues and no nonzero real eigenvectors.

Worked Example

Find the eigenvectors of the matrix below.

A = [[2, 1], [1, 2]]

1. Find the characteristic equation

Subtract λ from the diagonal and set the determinant to zero:

(2 − λ)(2 − λ) − (1 × 1) = 0
λ² − 4λ + 3 = 0
(λ − 3)(λ − 1) = 0
λ₁ = 3,   λ₂ = 1

2. Find an eigenvector for λ = 3

(A − 3I)v = 0
[[-1, 1], [1, -1]] [x, y]ᵀ = [0, 0]ᵀ
−x + y = 0, so y = x.
v₁ = (1, 1)

3. Find an eigenvector for λ = 1

(A − I)v = 0
[[1, 1], [1, 1]] [x, y]ᵀ = [0, 0]ᵀ
x + y = 0, so y = −x.
v₂ = (1, −1)

4. Verify the eigenvectors

A(1, 1)ᵀ = (3, 3)ᵀ = 3(1, 1)ᵀ
A(1, −1)ᵀ = (1, −1)ᵀ = 1(1, −1)ᵀ

Both vectors satisfy Av = λv, so both are valid eigenvectors.

Frequently Asked Questions

What is an eigenvector?

An eigenvector is a nonzero vector v that satisfies Av = λv for a scalar λ called its eigenvalue. Multiplication by the matrix changes the vector by a scalar factor without changing its direction, except that a negative factor reverses its direction.

How do you calculate eigenvectors of a 2×2 matrix?

First solve det(A - λI) = 0 to find the eigenvalues. Then, for each eigenvalue λ, solve the homogeneous system (A - λI)v = 0. Any nonzero solution is an eigenvector.

Can an eigenvector be multiplied by a number?

Yes. If v is an eigenvector, then kv is also an eigenvector for every nonzero scalar k. The zero vector is never considered an eigenvector.

What happens when an eigenvalue is repeated?

A repeated eigenvalue may have one or two linearly independent eigenvectors for a 2×2 matrix. If the matrix is a scalar multiple of the identity, every nonzero vector is an eigenvector. Otherwise, a real 2×2 matrix with a repeated eigenvalue has only one independent eigenvector direction.

Why might a matrix have no real eigenvectors?

A real matrix can have complex eigenvalues. For a real 2×2 matrix, a negative discriminant in the characteristic equation means its eigenvalues are complex, so there are no nonzero real eigenvectors.

Does this calculator support 3×3 matrices?

This calculator supports real 1×1 and 2×2 matrices. Larger matrices require solving higher-degree characteristic equations and finding the null space for each eigenvalue; use a dedicated higher-dimensional eigenvalue solver for those cases.