Find the eigen values and eigen vectors of following matrix 

  4     2    -2
-5     3     2
-2     4     1

To find the eigenvalues and eigenvectors of the given \(3 \times 3\) matrix:

\[
A = \begin{bmatrix}
4 & 2 & -2 \\
-5 & 3 & 2 \\
-2 & 4 & 1
\end{bmatrix}
\]

### Steps:
1. **Eigenvalues**:
   The eigenvalues \(\lambda\) satisfy the characteristic equation:
   \[
   \det(A - \lambda I) = 0
   \]
   Here, \(I\) is the identity matrix of the same size as \(A\).

   \[
   A - \lambda I =
   \begin{bmatrix}
   4 - \lambda & 2 & -2 \\
   -5 & 3 - \lambda & 2 \\
   -2 & 4 & 1 - \lambda
   \end{bmatrix}
   \]

   The determinant of \(A - \lambda I\) is calculated as:
   \[
   \det(A - \lambda I) = 
   \begin{vmatrix}
   4 - \lambda & 2 & -2 \\
   -5 & 3 - \lambda & 2 \\
   -2 & 4 & 1 - \lambda
   \end{vmatrix}
   \]

   Expanding this determinant results in a cubic equation in \(\lambda\).

2. **Eigenvectors**:
   For each eigenvalue \(\lambda\), solve the system \((A - \lambda I)\mathbf{x} = 0\) to find the eigenvector(s) \(\mathbf{x}\).

I will calculate the eigenvalues and eigenvectors explicitly. sympy as sp

# Define the matrix A
A = sp.Matrix([
    [4, 2, -2],
    [-5, 3, 2],
    [-2, 4, 1]
])

# Compute the eigenvalues and eigenvectors
eigen_data = A.eigenvects()

eigen_dataThe eigenvalues and corresponding eigenvectors of the matrix are as follows:

1. **Eigenvalue** \(\lambda = 1\):
   - Eigenvector: \[
   \mathbf{x}_1 = \begin{bmatrix}
   \frac{1}{2} \\
   \frac{1}{4} \\
   1
   \end{bmatrix}
   \]

2. **Eigenvalue** \(\lambda = 2\):
   - Eigenvector: \[
   \mathbf{x}_2 = \begin{bmatrix}
   \frac{1}{2} \\
   \frac{1}{2} \\
   1
   \end{bmatrix}
   \]

3. **Eigenvalue** \(\lambda = 5\):
   - Eigenvector: \[
   \mathbf{x}_3 = \begin{bmatrix}
   0 \\
   1 \\
   1
   \end{bmatrix}
   \]

These results show that the matrix has three distinct eigenvalues, each with a corresponding eigenvector.

---

Would you like detailed steps for how the eigenvalues and eigenvectors were calculated?  
Here are five questions you might want to explore next:

1. How is the characteristic polynomial derived in detail?
2. What does the geometric interpretation of eigenvectors and eigenvalues look like?
3. Can eigenvalues be used to diagonalize the matrix \(A\)?
4. What does the determinant of the matrix tell us about its eigenvalues?
5. How do eigenvalues relate to the stability of a system in applied problems?

**Tip:** Eigenvectors can be scaled by any non-zero scalar. They are typically normalized to make computations easier.

Find the eigenvalues and eigenvectors of the following matrix: 
 4  2 -2
 -5  3  2
 -2  4  1

Learn how to find the eigenvalues and eigenvectors of the matrix A = [4, 2, -2; -5, 3, 2; -2, 4, 1]. This problem includes key linear algebra concepts such as the characteristic equation, matrix operations, and solving for eigenvalues and eigenvectors. Suitable for undergraduate students.

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