Free Online Calculator

Eigenvectors Calculator

Name: Eigenvectors Calculator
Author: AskMathAI

Calculate matrix eigenvectors step-by-step with our advanced linear algebra calculator. Perfect for finding eigenvectors for any matrix size with detailed null space solutions.

Eigenvectors Calculator

Enter a matrix and calculate its eigenvectors with step-by-step solutions

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Master Eigenvectors with Our Advanced Calculator

Our eigenvectors calculator is designed to help students, teachers, and professionals calculate matrix eigenvectors efficiently. Whether you're working on linear algebra homework, preparing foruniversity mathematics exams, or tackling engineering problems, this tool provides comprehensive step-by-step solutions that enhance your understanding of matrix operations.

The eigenvectors calculator computes the eigenvectors of matrices by solving the null space of $(A - \lambda I)v = 0$ for each eigenvalue $\lambda$ . Our linear algebra calculator is particularly useful for university linear algebra courses and engineering applications, where eigenvectors are used in diagonalization, principal component analysis, quantum mechanics, and solving systems of differential equations.

Perfect for high school linear algebra students learning matrix properties, university studentsin advanced mathematics courses, engineering students applying eigenvectors to real-world problems, andprofessionals who need quick mathematical solutions. The eigenvectors calculator provides not just answers, but detailed explanations that help you understand the underlying concepts and improve your mathematical skills.

Eigenvectors Properties and Applications

Property	Formula	Description	Application
Eigenvector Definition	$(A - \lambda I)v = 0$	Fundamental equation for eigenvectors	Finding eigenvectors
Null Space	$v \in \text{Null}(A - \lambda I)$	Eigenvectors belong to null space	Linear independence
Eigenvalue-Eigenvector	$Av = \lambda v$	Direct relationship	Matrix transformations
Linear Independence	$\{v_1, v_2, ...\}$ linearly independent	Different eigenvalues have independent eigenvectors	Basis formation
Diagonalization	$A = PDP^{-1}$ where $P$ contains eigenvectors	Eigenvectors form transformation matrix	Matrix simplification

Common Mistakes to Avoid

Zero Vector

The zero vector $\mathbf{0}$ is NOT an eigenvector. Eigenvectors must be non-zero vectors that satisfy $(A - \lambda I)\mathbf{v} = \mathbf{0}$ .

Multiplicity

For repeated eigenvalues, you may need multiple linearly independent eigenvectors. Check the geometric multiplicity vs algebraic multiplicity.

Normalization

Eigenvectors are not unique - any scalar multiple is also an eigenvector. Consider normalizing to unit length for consistency.

How to Calculate Eigenvectors

Eigenvector calculation involves solving the null space of the matrix (A - λI) for each eigenvalue λ.

$(A - \lambda I)\mathbf{v} = \mathbf{0}$

This is the fundamental equation for eigenvectors. The eigenvectors $\mathbf{v}$ are the non-zero vectors that satisfy this homogeneous system of equations.

Calculation Steps

Find Eigenvalues

Solve det(A - λI) = 0

Form A - λI

Subtract λ from diagonal for each eigenvalue

Solve Null Space

Find non-zero solutions to (A - λI)v = 0

Verify Results

Check Av = λv for each eigenvector

Key Eigenvector Properties

$A\mathbf{v} = \lambda\mathbf{v}$

$\mathbf{v} \neq \mathbf{0}$

$c\mathbf{v} \text{ is also eigenvector}$

Examples

2×2 Matrix

$A = [[3, 1], [0, 2]]$

Steps:

Find eigenvalues λ₁=3, λ₂=2
Solve (A-λI)v = 0 for each λ
Find null space solutions

v₁ = [1, 0], v₂ = [-1, 1]

Symmetric Matrix

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

Steps:

Find eigenvalues λ₁=3, λ₂=1
Form A-λI for each eigenvalue
Solve homogeneous system

v₁ = [1, 1], v₂ = [1, -1]

Diagonal Matrix

$A = [[4, 0], [0, -1]]$

Steps:

Eigenvalues are diagonal elements
Standard basis vectors are eigenvectors
No calculation needed

v₁ = [1, 0], v₂ = [0, 1]

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Frequently Asked Questions

What are eigenvectors?

Eigenvectors are non-zero vectors $v$ that satisfy the equation $Av = \lambda v$ for some scalar $\lambda$ (the eigenvalue). They represent directions that are preserved under the linear transformation represented by matrix $A$ .

How do I calculate eigenvectors?

To calculate eigenvectors, first find the eigenvalues by solving $\det(A - \lambda I) = 0$ . Then for each eigenvalue $\lambda$ , solve the homogeneous system $(A - \lambda I)\mathbf{v} = \mathbf{0}$ to find the corresponding eigenvectors.

Can eigenvectors be zero?

No, eigenvectors cannot be the zero vector. By definition, eigenvectors must be non-zero vectors. The zero vector satisfies $A\mathbf{0} = \lambda\mathbf{0}$ for any $\lambda$ , but it is not considered an eigenvector.

Are eigenvectors unique?

Eigenvectors are not unique - any scalar multiple of an eigenvector is also an eigenvector. For example, if $\mathbf{v}$ is an eigenvector, then $c\mathbf{v}$ is also an eigenvector for any non-zero scalar $c$ .

What is the relationship between eigenvalues and eigenvectors?

Each eigenvalue $\lambda$ has corresponding eigenvectors that satisfy $A\mathbf{v} = \lambda\mathbf{v}$ . The number of linearly independent eigenvectors for an eigenvalue is called its geometric multiplicity, which cannot exceed the algebraic multiplicity of the eigenvalue.

Is this calculator free to use?

Yes, our eigenvectors calculator is completely free to use with no limitations. You can calculate eigenvectors of as many matrices as you need.

Eigenvectors Calculator