Free Eigenvalue Calculator

Calculate eigenvalues and eigenvectors for 2x2 and 3x3 matrices instantly

Eigenvalues and Eigenvectors

What are Eigenvalues?

Eigenvalues are special numbers associated with a square matrix that provide important information about the matrix's properties. For a given square matrix A, an eigenvalue λ is a scalar value that satisfies the equation:

A·v = λ·v

Where v is a non-zero vector called the eigenvector corresponding to the eigenvalue λ.

How to Calculate Eigenvalues

To find the eigenvalues of a matrix:

1. Start with the equation: A·v = λ·v
2. Rewrite as: (A - λI)·v = 0, where I is the identity matrix
3. For non-zero solutions, the determinant must be zero: det(A - λI) = 0
4. Solve this characteristic equation to find the eigenvalues λ

Applications of Eigenvalues

Eigenvalues have numerous applications in various fields:

• Physics: Vibration analysis, quantum mechanics
• Engineering: Stability analysis, control systems
• Computer Science: Google's PageRank algorithm, image processing
• Statistics: Principal Component Analysis (PCA)
• Economics: Input-output models, portfolio optimization

Using This Calculator

Our free eigenvalue calculator makes it easy to find eigenvalues and eigenvectors for 2x2 and 3x3 matrices:

1. Select the matrix size (2x2 or 3x3)
2. Enter the values of your matrix in the input fields
3. Click "Calculate Eigenvalues" to see the results
4. Use the "Reset" button to clear all inputs and results