Eigenvalue and Eigenvector Calculator

Calculate eigenvalues and eigenvectors for 2×2 and 3×3 matrices with step-by-step solutions

Matrix Input

Enter real numbers for matrix elements

Results

5.000
Trace (tr(A))
-2.000
Determinant (|A|)

Eigenvalues

λ1 =5.372281
λ2 =-0.372281

Eigenvectors

v1 =[-2.000, -4.372]
v2 =[-2.000, 1.372]

Characteristic Polynomial

det(A - λI) = λ² - 5.000λ + -2.000 = 0

Formulas Used

For 2×2 matrices:

λ = (tr(A) ± √(tr(A)² - 4|A|)) / 2

Eigenvector equation: (A - λI)v = 0

Trace: Sum of diagonal elements

Determinant (2×2): ad - bc

Example Calculation

2×2 Matrix Example

Matrix A: [[3, 1], [0, 2]]

Trace: 3 + 2 = 5

Determinant: 3×2 - 1×0 = 6

Characteristic equation: λ² - 5λ + 6 = 0

Solution

Eigenvalues: λ₁ = 3, λ₂ = 2

Eigenvector for λ₁ = 3: [1, 0]

Eigenvector for λ₂ = 2: [-1, 1]

Key Concepts

λ

Eigenvalue

Scalar that scales eigenvector

v

Eigenvector

Vector that doesn't change direction

tr

Trace

Sum of diagonal elements

Applications

Principal Component Analysis (PCA)

Stability analysis of systems

Google PageRank algorithm

Quantum mechanics

Machine learning algorithms

Understanding Eigenvalues and Eigenvectors

What are Eigenvalues and Eigenvectors?

An eigenvector of a square matrix A is a non-zero vector v that, when A is multiplied by v, yields a scalar multiple of v. The scalar multiple is called the eigenvalue λ.

A × v = λ × v

Characteristic Equation

To find eigenvalues, we solve the characteristic equation:

det(A - λI) = 0

For 2×2 Matrices

For a 2×2 matrix, eigenvalues can be found using the quadratic formula:

λ = (tr(A) ± √(tr(A)² - 4|A|)) / 2

Properties

  • • Sum of eigenvalues equals the trace
  • • Product of eigenvalues equals the determinant
  • • Eigenvectors are linearly independent
  • • Complex eigenvalues occur in conjugate pairs

Note: For 3×3 matrices, finding exact eigenvalues requires solving cubic equations, which can be computationally intensive.