Eigenvalues and Eigenvectors Calculator

Choose any matrix order 2x2, 3x3, 4x4, 5x5, put values, and this eigenvalues and eigenvectors calculator will instantly determine the answer within steps shown.

Choose the size of the matrix:

Eigenvalues and Eigenvectors Calculator

This calculator helps calculate eigenvalues and eigenvectors (eigenspace) of a given square matrix and provides step-by-step solutions. It supports 2×2, 3×3, 4×4, and 5×5 matrices.

What are Eigenvalues & Eigenvectors
Formula to Find Eigenvalue & Eigenvector
Relation between Eigenvalue & Eigenvector
Step-by-Step Example of Calculating Eigenpairs
How to Use this Eigenvalue and Eigenvector Calculator
Eigenvalue and Eigenvector Decomposition of a Matrix
Importance of Eigenvalues and Eigenvectors

What are Eigenvalues & Eigenvectors?

Eigenvalue

An eigenvalue is a scalar (λ) associated with a square matrix A. It represents how a linear transformation scales a corresponding eigenvector without changing its direction.

Eigenvector

An eigenvector is a non-zero vector that, when transformed by the matrix A, changes only in magnitude (scaled by its eigenvalue) but not in direction. Mathematically:

Av = λv

Formulas to Calculate Eigenvalues & Eigenvectors

Eigenvalues

The characteristic equation is:

det(A − λI) = 0

Eigenvectors

The vector equation is:

(A − λI)v = 0

A → square matrix
λ → eigenvalue
I → identity matrix
v → eigenvector

Relation Between Eigenvalue & Eigenvector

The fundamental relationship is:

Av = λv

Here, λ represents the scaling factor, and v is the vector that is only scaled and not rotated. This equation can also be written as (A − λI)v = 0 to solve for eigenpairs.

Step-by-Step Example

Consider the matrix:

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

Find the Eigenvalues

Create the characteristic equation: det(A − λI) = 0
Compute:
```
A − λI = [[2−λ, 1], [1, 2−λ]]
```
Determinant: (2−λ)(2−λ) − 1 = 0 → (2−λ)² − 1 = 0
Solve: 2−λ = ±1 → λ₁ = 3, λ₂ = 1

Find Eigenvectors

Eigenvector for λ = 3

A − 3I = [[-1, 1], [1, -1]]

Solve (A − λI)v = 0 → -x + y = 0 → y = x

Eigenvector: v₁ = [1, 1]

Eigenvector for λ = 1

A − I = [[1, 1], [1, 1]]

Solve (A − λI)v = 0 → x + y = 0 → y = -x

Eigenvector: v₂ = [1, -1]

How to Use the Calculator

Choose matrix size (2×2, 3×3, 4×4, or 5×5).
Enter the matrix elements.
Click “CALCULATE” to get eigenvalues and eigenvectors instantly.

Eigenvalue and Eigenvector Decomposition

Matrix A can be decomposed as:

A = PDP⁻¹

P → Matrix of eigenvectors
D → Diagonal matrix of eigenvalues
P⁻¹ → Inverse of P

For symmetric matrices: A = PDPᵀ

Importance of Eigenvalues and Eigenvectors

System dynamics and stability analysis
Quantum mechanics state analysis
Principal Component Analysis (PCA) in data science
Vibrations and mechanical systems analysis
Markov chains and steady-state probability distributions
Matrix diagonalization for simplified computations
Graph theory and network analysis

FAQs

What does it mean when a matrix has multiple eigenvalues?

It implies the matrix acts on vectors in multiple ways. Repeated eigenvalues may have one or more corresponding eigenvectors.

Can this calculator handle 3×3 matrices?

Yes, it supports 2×2, 3×3, 4×4, and 5×5 matrices.

How to find eigenvectors once eigenvalues are known?

Use the equation (A − λI)v = 0
Substitute the known eigenvalue λ
Solve for v
Any non-zero solution is an eigenvector
Repeat for each eigenvalue

Are there other names for eigenvalues?

Latent roots
Characteristic roots
Characteristic values
Proper values

References

Wikipedia: Eigenvalues and eigenvectors
Libretext Mathematics: Eigenvalues and Eigenvectors of a Matrix
Marcus, M. and Minc, H. Introduction to Linear Algebra, Dover, 1988.
Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. Numerical Recipes in FORTRAN: The Art of Scientific Computing, Cambridge University Press, 1992.