Eigenvalues and Eigenvectors Calculator

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?

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: det(A − λI) = 0

Eigenvectors: (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: det(A − λI) = 0 → (2−λ)^2 − 1 = 0 → λ₁ = 3, λ₂ = 1

Eigenvector for λ = 3: v₁ = [1, 1]

Eigenvector for λ = 1: v₂ = [1, −1]

How to Use the Calculator

Choose matrix size (2×2, 3×3, 4×4, or 5×5), enter the matrix elements, and click "CALCULATE" to get eigenvalues and eigenvectors instantly.

Eigenvalue and Eigenvector Decomposition

Matrix A can be decomposed as A = PDP⁻¹, where P is the matrix of eigenvectors, D is the diagonal matrix of eigenvalues, and P⁻¹ is the inverse of P. For symmetric matrices: A = PDPᵀ.

Importance of Eigenvalues and Eigenvectors

Applications include system dynamics and stability analysis, quantum mechanics state analysis, PCA in data science, vibrations and mechanical systems analysis, Markov chains, matrix diagonalization, and graph/network analysis.

FAQs

Multiple eigenvalues may have one or more corresponding eigenvectors.

Yes, the calculator supports 2×2, 3×3, 4×4, and 5×5 matrices.

To find eigenvectors once eigenvalues are known, solve (A − λI)v = 0 for each eigenvalue.

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. et al., Numerical Recipes in FORTRAN, Cambridge University Press, 1992.