Eigenvectors and Eigenvalues

What are Eigenvectors and Eigenvalues?

Eigenvectors and eigenvalues are fundamental concepts in linear algebra with wide-ranging applications in physics, engineering, data science, and more. For a square matrix A, an eigenvector is a non-zero vector v that, when multiplied by A, yields a scalar multiple of itself. This scalar is called the eigenvalue.

The Eigenvalue Equation

Mathematically, we express this as:

\[A v = \lambda v\]

Where:

• A is a square matrix
• v is an eigenvector (non-zero)
• \(\lambda\) (lambda) is the corresponding eigenvalue

Calculating Eigenvectors and Eigenvalues

For an n x n matrix A, we follow these steps:

1. Find the characteristic equation: \(\det(A - \lambda I) = 0\)
2. Solve for \(\lambda\) to get the eigenvalues
3. For each \(\lambda\), solve \((A - \lambda I)v = 0\) to find the corresponding eigenvector

Example Calculation

Let's calculate the eigenvectors and eigenvalues for the matrix:

\[A = \begin{bmatrix} 3 & -2 \\ 1 & 0 \end{bmatrix}\]

1. Characteristic equation: \[\det(A - \lambda I) = \begin{vmatrix} 3-\lambda & -2 \\ 1 & -\lambda \end{vmatrix} = (3-\lambda)(-\lambda) - (-2)(1) = \lambda^2 - 3\lambda + 2 = 0\]
2. Solve for \(\lambda\): \[\lambda = \frac{3 \pm \sqrt{9 - 8}}{2} = \frac{3 \pm 1}{2}\] \[\lambda_1 = 2, \lambda_2 = 1\]
3. Find eigenvectors: For \(\lambda_1 = 2\): \[(A - 2I)v = \begin{bmatrix} 1 & -2 \\ 1 & -2 \end{bmatrix}v = 0\] \[v_1 = \begin{bmatrix} 2 \\ 1 \end{bmatrix}\] For \(\lambda_2 = 1\): \[(A - I)v = \begin{bmatrix} 2 & -2 \\ 1 & -1 \end{bmatrix}v = 0\] \[v_2 = \begin{bmatrix} 1 \\ 1 \end{bmatrix}\]

Visual Representation

This diagram shows the eigenvectors v₁ (blue) and v₂ (red) for the example matrix.