# Eigenvalue Calculator

Calculate eigenvalues and eigenvectors of 2x2 and 3x3 matrices. Shows the characteristic polynomial, trace, and determinant. Free online eigenvalue calculator.

## What this calculates

Find the eigenvalues and eigenvectors of a 2x2 or 3x3 matrix. This calculator solves the characteristic polynomial, computes the trace and determinant, and displays the corresponding eigenvectors.

## Inputs

- **Matrix Size** — options: 2 x 2, 3 x 3 — Choose the dimension of your square matrix.
- **a₁₁ (Row 1, Col 1)**
- **a₁₂ (Row 1, Col 2)**
- **a₁₃ (Row 1, Col 3)** — Only used for 3x3 matrices.
- **a₂₁ (Row 2, Col 1)**
- **a₂₂ (Row 2, Col 2)**
- **a₂₃ (Row 2, Col 3)** — Only used for 3x3 matrices.
- **a₃₁ (Row 3, Col 1)** — Only used for 3x3 matrices.
- **a₃₂ (Row 3, Col 2)** — Only used for 3x3 matrices.
- **a₃₃ (Row 3, Col 3)** — Only used for 3x3 matrices.

## Outputs

- **Eigenvalues** — formatted as text — The eigenvalues of the matrix.
- **Eigenvectors** — formatted as text — The eigenvectors corresponding to each eigenvalue.
- **Characteristic Polynomial** — formatted as text — The polynomial det(A - lambda I) = 0.
- **Trace** — Sum of diagonal elements (equals sum of eigenvalues).
- **Determinant** — Product of eigenvalues.

## Details

Eigenvalues and eigenvectors are fundamental concepts in linear algebra. An eigenvector of a matrix A is a non-zero vector v such that Av = lambda * v, where lambda (the eigenvalue) is a scalar.

**Finding Eigenvalues:**

Solve the characteristic equation: det(A - lambda * I) = 0. For a 2x2 matrix, this gives a quadratic. For a 3x3, a cubic.

**Finding Eigenvectors:**

For each eigenvalue lambda, solve the system (A - lambda * I)v = 0. The solution space (null space) gives the eigenvector(s) for that eigenvalue.

**Key Properties:**

- The trace (sum of diagonal elements) equals the sum of eigenvalues
- The determinant equals the product of eigenvalues
- Symmetric matrices always have real eigenvalues
- Eigenvectors for distinct eigenvalues are linearly independent

**Applications:**

Eigenvalues and eigenvectors appear everywhere: principal component analysis (PCA) in data science, vibration analysis in engineering, Google's PageRank algorithm, quantum mechanics (observable operators), stability analysis of dynamic systems, and image compression.

## Frequently Asked Questions

**Q: What is an eigenvalue?**

A: An eigenvalue is a scalar lambda such that for a square matrix A, there exists a non-zero vector v where Av = lambda * v. In other words, multiplying the matrix by this special vector just scales it, without changing its direction. Every n x n matrix has exactly n eigenvalues (counting multiplicity and complex values).

**Q: What is the characteristic polynomial?**

A: The characteristic polynomial is det(A - lambda * I) = 0, where I is the identity matrix. For a 2x2 matrix, it is a quadratic equation in lambda. For a 3x3 matrix, it is a cubic. The roots of this polynomial are the eigenvalues of the matrix.

**Q: Can eigenvalues be complex numbers?**

A: Yes. Non-symmetric real matrices can have complex eigenvalues, which always come in conjugate pairs. For example, a 2D rotation matrix has complex eigenvalues because rotation does not leave any real direction unchanged. Symmetric matrices, however, always have real eigenvalues.

**Q: Where are eigenvalues used in practice?**

A: Eigenvalues are used in Google's PageRank (dominant eigenvector of the web graph), principal component analysis for data reduction, vibration analysis of structures and circuits, quantum mechanics (energy levels are eigenvalues of the Hamiltonian), and stability analysis of differential equations and control systems.

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Source: https://vastcalc.com/calculators/math/eigenvalue-eigenvector
Category: Math
Last updated: 2026-04-08