Eigenvector Calculator

Calculate the eigenvalues and corresponding eigenvectors for a 2x2 matrix.

A =

Results:

Understanding Eigenvectors

While an eigenvalue (λ) tells you how much a vector is stretched or shrunk during a linear transformation, an eigenvector (v) tells you the direction of that stretch. For a matrix transformation A, the eigenvectors are the lines that do not change direction after the transformation is applied.

(A - λI)v = 0

How to Find Eigenvectors (2x2)

  1. Find Eigenvalues: First, solve the characteristic equation det(A - λI) = 0 to get the values of λ.
  2. Substitute: For each λ, substitute it back into the equation (A - λI)v = 0.
  3. Solve the System: This gives you a system of linear equations. Since the matrix (A - λI) is singular, the equations will be dependent. You need to find a non-zero vector that satisfies them.

Example

Consider the matrix A = [[2, 1], [1, 2]].

The eigenvalues are λ₁ = 3 and λ₂ = 1.

For λ₁ = 3:

(A - 3I)v = [[-1, 1], [1, -1]] * [x, y] = [0, 0]

This implies -x + y = 0, or x = y. So, any vector where x=y is an eigenvector, e.g., [1, 1].

For λ₂ = 1:

(A - 1I)v = [[1, 1], [1, 1]] * [x, y] = [0, 0]

This implies x + y = 0, or x = -y. So, any vector where x=-y is an eigenvector, e.g., [1, -1].

Applications

Eigenvectors are fundamental in physics (quantum states, rotation axes), computer graphics (image compression, PCA), engineering (stress tensor principal directions), and even Google's PageRank algorithm.