To find **eigenvalues** and **eigenvectors** of a square matrix, you need to follow these steps:

### Steps to Find Eigenvalues:
1. **Start with a square matrix \( A \):**
   The matrix must be square (e.g., \( 2 \times 2 \), \( 3 \times 3 \)) for eigenvalues and eigenvectors to exist.

2. **Set up the characteristic equation:**
   The characteristic equation is given by:

   \[
   \text{det}(A - \lambda I) = 0
   \]

   Here:
   - \( A \) is your original matrix.
   - \( \lambda \) is the eigenvalue.
   - \( I \) is the identity matrix of the same size as \( A \).

3. **Solve for \( \lambda \) (the eigenvalues):**
   Find the values of \( \lambda \) that satisfy the characteristic equation. These are the eigenvalues.

### Steps to Find Eigenvectors:
1. **Substitute each eigenvalue \( \lambda \) into the equation:**

   \[
   (A - \lambda I) \mathbf{v} = 0
   \]

   Here:
   - \( \mathbf{v} \) is the eigenvector corresponding to the eigenvalue \( \lambda \).

2. **Solve the system of linear equations:**
   For each \( \lambda \), solve the matrix equation \( (A - \lambda I) \mathbf{v} = 0 \) to find the corresponding eigenvector \( \mathbf{v} \). You will usually find the eigenvector up to a constant multiple, as any scalar multiple of an eigenvector is also an eigenvector.

### Example:
Let's find the eigenvalues and eigenvectors for a simple \( 2 \times 2 \) matrix:

\[
A = \begin{pmatrix}
4 & 1 \\
2 & 3
\end{pmatrix}
\]

1. Set up the characteristic equation:

   \[
   \text{det}(A - \lambda I) = 0
   \]

   \[
   \text{det}\left(\begin{pmatrix} 4 - \lambda & 1 \\ 2 & 3 - \lambda \end{pmatrix}\right) = 0
   \]

2. Compute the determinant:

   \[
   (4 - \lambda)(3 - \lambda) - (1 \cdot 2) = 0
   \]

   \[
   \lambda^2 - 7\lambda + 10 = 0
   \]

   Solve this quadratic equation:

   \[
   \lambda = 5 \quad \text{or} \quad \lambda = 2
   \]

3. For each eigenvalue, solve for the eigenvectors. Let's start with \( \lambda = 5 \):

   \[
   (A - 5I) \mathbf{v} = 0
   \]

   \[
   \begin{pmatrix} -1 & 1 \\ 2 & -2 \end{pmatrix} \mathbf{v} = 0
   \]

   Solve this system, and you'll find \( \mathbf{v} = \begin{pmatrix} 1 \\ 1 \end{pmatrix} \). This is an eigenvector corresponding to \( \lambda = 5 \).

4. Similarly, solve for \( \lambda = 2 \) to find another eigenvector.

Would you like a more detailed explanation or another example?

### Related Questions:
1. What are the properties of eigenvalues and eigenvectors?
2. How are eigenvalues related to the trace and determinant of a matrix?
3. Can a matrix have complex eigenvalues?
4. How do you find eigenvalues for larger matrices (e.g., \( 3 \times 3 \))?
5. What are the geometric interpretations of eigenvalues and eigenvectors?

### Tip:
For larger matrices or more complicated calculations, you can use software tools like MATLAB, NumPy (Python), or even graphing calculators to compute eigenvalues and eigenvectors efficiently.

Learn how to find eigenvalues and eigenvectors of a square matrix using the characteristic equation. This detailed guide covers the necessary formulas and step-by-step solutions, including an example with a 2x2 matrix. Suitable for undergraduate students studying linear algebra.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation