Find Eigen values and Eigen vectors of A = [[2, - 2, 2], [1, 1, 1], [1, 3, - 1]]

To find the eigenvalues and eigenvectors of the matrix \( A = \begin{bmatrix} 2 & -2 & 2 \\ 1 & 1 & 1 \\ 1 & 3 & -1 \end{bmatrix} \), we need to follow these steps:

### Step 1: Find the Eigenvalues
Eigenvalues \( \lambda \) are found by solving the characteristic equation:
\[
\det(A - \lambda I) = 0
\]
where \( I \) is the identity matrix.

The matrix \( A - \lambda I \) is:
\[
A - \lambda I = \begin{bmatrix} 
2 - \lambda & -2 & 2 \\
1 & 1 - \lambda & 1 \\
1 & 3 & -1 - \lambda 
\end{bmatrix}
\]

Now, compute the determinant of this matrix to find the characteristic polynomial.

### Step 2: Find the Eigenvectors
Once we find the eigenvalues, for each eigenvalue \( \lambda \), we need to solve the system:
\[
(A - \lambda I) \mathbf{v} = 0
\]
where \( \mathbf{v} \) is the eigenvector corresponding to the eigenvalue \( \lambda \).

Let me calculate these steps.The eigenvalues and corresponding eigenvectors of the matrix \( A \) are as follows:

### Eigenvalues:
\[
\lambda_1 = 2.00, \quad \lambda_2 = 2.00, \quad \lambda_3 = -2.00
\]

### Eigenvectors:
- For \( \lambda_1 = 2.00 \):
  \[
  \mathbf{v}_1 = \begin{bmatrix} 2.166 \times 10^{-8} \\ -0.7071 \\ -0.7071 \end{bmatrix}
  \]
  
- For \( \lambda_2 = 2.00 \):
  \[
  \mathbf{v}_2 = \begin{bmatrix} -2.166 \times 10^{-8} \\ -0.7071 \\ -0.7071 \end{bmatrix}
  \]

- For \( \lambda_3 = -2.00 \):
  \[
  \mathbf{v}_3 = \begin{bmatrix} -0.4924 \\ -0.1231 \\ 0.8616 \end{bmatrix}
  \]

Would you like more details on any of these steps or clarifications?

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Here are 5 follow-up questions you might consider:
1. How can I use eigenvectors in practical applications like physics or engineering?
2. What is the geometric significance of eigenvectors and eigenvalues?
3. How do degenerate eigenvalues (repeated ones) affect the eigenvectors?
4. What is the process to diagonalize a matrix using its eigenvalues and eigenvectors?
5. How do I compute eigenvalues for larger matrices?

**Tip:** Eigenvalues can reveal important properties of systems, like stability in mechanical systems or natural frequencies in structures.

Explore how to find eigenvalues and eigenvectors of the matrix A = [[2, -2, 2], [1, 1, 1], [1, 3, -1]]. This solution walks through calculating the determinant, solving for eigenvalues, and finding the corresponding eigenvectors. Suitable for undergraduate linear algebra students.

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