Eigenvectors and Eigenvalues for 3x3 Matrices

What are Eigenvectors and Eigenvalues?

Eigenvectors and eigenvalues are fundamental concepts in linear algebra with wide-ranging applications in physics, engineering, and data science. 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 3x3 matrix
• v is an eigenvector (non-zero)
• \(\lambda\) (lambda) is the corresponding eigenvalue

Calculating Eigenvectors and Eigenvalues for 3x3 Matrices

For a 3x3 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:

\[A = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix}\]

1. Characteristic equation: \[\det(A - \lambda I) = \begin{vmatrix} 1-\lambda & 2 & 3 \\ 4 & 5-\lambda & 6 \\ 7 & 8 & 9-\lambda \end{vmatrix} = -\lambda^3 + 15\lambda^2 - 18\lambda = 0\]
2. Solve for \(\lambda\): \[\lambda(\lambda^2 - 15\lambda + 18) = 0\] \[\lambda_1 = 0, \lambda_2 = \frac{15 + \sqrt{33}}{2}, \lambda_3 = \frac{15 - \sqrt{33}}{2}\]
3. Find eigenvectors: For \(\lambda_1 = 0\): \[(A - 0I)v = Av = 0\] \[v_1 = \begin{bmatrix} 1 \\ -2 \\ 1 \end{bmatrix}\] For \(\lambda_2 = \frac{15 + \sqrt{33}}{2}\): \[(A - \lambda_2 I)v = 0\] \[v_2 \approx \begin{bmatrix} 0.464 \\ 0.570 \\ 0.677 \end{bmatrix}\] For \(\lambda_3 = \frac{15 - \sqrt{33}}{2}\): \[(A - \lambda_3 I)v = 0\] \[v_3 \approx \begin{bmatrix} -0.884 \\ 0.240 \\ 0.405 \end{bmatrix}\]

Visual Representation

This diagram shows the eigenvectors v₁ (blue), v₂ (red), and v₃ (green) for the example 3x3 matrix, projected onto a 2D plane.