Find the eigenvalues and eigen vectors of A = (13,-4,2,  -4,13,-2,    2, -2, 10)

To find the eigenvalues and eigenvectors of the matrix \( A \), follow these steps:

### Matrix \( A \):
\[
A = \begin{pmatrix}
13 & -4 & 2 \\
-4 & 13 & -2 \\
2 & -2 & 10
\end{pmatrix}
\]

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

The characteristic equation is derived by computing the determinant:
\[
\det(A - \lambda I) = \det \begin{pmatrix}
13 - \lambda & -4 & 2 \\
-4 & 13 - \lambda & -2 \\
2 & -2 & 10 - \lambda
\end{pmatrix}
\]

### Step 2: Find the Eigenvectors
For each eigenvalue \( \lambda \), solve the equation:
\[
(A - \lambda I)v = 0
\]
to find the eigenvectors \( v \).

Let me compute these values for you.import numpy as np
from sympy import symbols, Matrix, solve

# Define the matrix A
A = Matrix([
    [13, -4, 2],
    [-4, 13, -2],
    [2, -2, 10]
])

# Define the eigenvalue variable
lambda_ = symbols('lambda')

# Compute the characteristic polynomial
char_poly = A.charpoly(lambda_).as_expr()

# Solve for eigenvalues
eigenvalues = solve(char_poly, lambda_)

# Compute eigenvectors for each eigenvalue
eigenvectors = {eig: (A - eig * Matrix.eye(3)).nullspace() for eig in eigenvalues}

char_poly, eigenvalues, eigenvectors(lambda**3 - 36*lambda**2 + 405*lambda - 1458,
 [9, 18],
 {9: [Matrix([
   [1],
   [1],
   [0]]),
   Matrix([
   [-1/2],
   [   0],
   [   1]])],
  18: [Matrix([
   [ 2],
   [-2],
   [ 1]])]})(lambda**3 - 36*lambda**2 + 405*lambda - 1458,
 [9, 18],
 {9: [Matrix([
   [1],
   [1],
   [0]]),
   Matrix([
   [-1/2],
   [   0],
   [   1]])],
  18: [Matrix([
   [ 2],
   [-2],
   [ 1]])]})success### Results:

1. **Characteristic Polynomial:**
   \[
   \lambda^3 - 36\lambda^2 + 405\lambda - 1458 = 0
   \]

2. **Eigenvalues:**
   \[
   \lambda_1 = 9 \quad \text{(with multiplicity 2)}, \quad \lambda_2 = 18
   \]

3. **Eigenvectors:**

   - For \( \lambda = 9 \):
     \[
     v_1 = \begin{pmatrix} 1 \\ 1 \\ 0 \end{pmatrix}, \quad v_2 = \begin{pmatrix} -\frac{1}{2} \\ 0 \\ 1 \end{pmatrix}
     \]

   - For \( \lambda = 18 \):
     \[
     v_3 = \begin{pmatrix} 2 \\ -2 \\ 1 \end{pmatrix}
     \]

These eigenvectors are normalized representations of the null space solutions corresponding to each eigenvalue.

Let me know if you need explanations on the steps or want to discuss eigenvalue properties!

---

### Related Questions:
1. How are eigenvalues related to the trace and determinant of a matrix?
2. What is the geometric significance of eigenvectors?
3. How do repeated eigenvalues affect diagonalization?
4. Can this matrix \( A \) be diagonalized? If so, how?
5. What role do eigenvalues play in solving systems of differential equations?

### Tip:
Always verify the orthogonality of eigenvectors for symmetric matrices—they are guaranteed to be orthogonal!

Find the eigenvalues and eigenvectors of A = (13, -4, 2, -4, 13, -2, 2, -2, 10).

Learn how to find the eigenvalues and eigenvectors of the matrix A = (13, -4, 2, -4, 13, -2, 2, -2, 10). This problem explores key linear algebra concepts, including the characteristic polynomial and eigenvector equations. Suitable for advanced high school or undergraduate students.

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