1 3 3
1 4 3
-1 3 4
Find eigen values and vectors

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

Given matrix:
\[
A = \begin{pmatrix}
1 & 3 & 3 \\
1 & 4 & 3 \\
-1 & 3 & 4
\end{pmatrix}
\]

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

1. Subtract \( \lambda \) from the diagonal elements of \( A \):
\[
A - \lambda I = \begin{pmatrix}
1-\lambda & 3 & 3 \\
1 & 4-\lambda & 3 \\
-1 & 3 & 4-\lambda
\end{pmatrix}
\]

2. Calculate the determinant:
\[
\text{det}(A - \lambda I) = \text{det} \begin{pmatrix}
1-\lambda & 3 & 3 \\
1 & 4-\lambda & 3 \\
-1 & 3 & 4-\lambda
\end{pmatrix}
\]

Expanding this determinant:
\[
(1-\lambda) \left( (4-\lambda)(4-\lambda) - 3 \times 3 \right) - 3 \left( 1(4-\lambda) - 3(-1) \right) + 3 \left( 1 \times 3 - 1 \times (-1) \right)
\]
Simplify and solve for \( \lambda \).

### Step 2: Find the Eigenvectors
For each eigenvalue \( \lambda_i \), solve:
\[
(A - \lambda_i I) \mathbf{v} = 0
\]
Where \( \mathbf{v} \) is the eigenvector associated with \( \lambda_i \).

### Calculation:
Let me calculate the eigenvalues and eigenvectors.### Eigenvalues:
The eigenvalues of the matrix \( A \) are approximately:
\[
\lambda_1 = 1 + 2.95 \times 10^{-8}i, \quad \lambda_2 = 1 - 2.95 \times 10^{-8}i, \quad \lambda_3 = 7
\]
Here, \( \lambda_1 \) and \( \lambda_2 \) are complex conjugates with very small imaginary parts, so they are almost real.

### Eigenvectors:
The corresponding eigenvectors are:

For \( \lambda_1 = 1 + 2.95 \times 10^{-8}i \):
\[
\mathbf{v}_1 \approx \begin{pmatrix} -1.57 \times 10^{-16} - 2.09 \times 10^{-8}i \\ -0.7071 \\ 0.7071 \end{pmatrix}
\]

For \( \lambda_2 = 1 - 2.95 \times 10^{-8}i \):
\[
\mathbf{v}_2 \approx \begin{pmatrix} -1.57 \times 10^{-16} + 2.09 \times 10^{-8}i \\ -0.7071 \\ 0.7071 \end{pmatrix}
\]

For \( \lambda_3 = 7 \):
\[
\mathbf{v}_3 \approx \begin{pmatrix} 0.5721 \\ 0.6674 \\ 0.4767 \end{pmatrix}
\]

These eigenvectors are normalized (each has a magnitude of 1).

Would you like further details on the steps, or do you have any questions?

### Related Questions:
1. What is the significance of eigenvalues in a matrix?
2. How are eigenvectors used in applications such as Principal Component Analysis (PCA)?
3. Can you explain how the characteristic polynomial is derived from a matrix?
4. What are some real-world applications of eigenvalues and eigenvectors?
5. How do complex eigenvalues affect the stability of a system?
6. What is the geometric interpretation of eigenvectors and eigenvalues?
7. How can you verify if a vector is an eigenvector of a matrix?
8. What happens if a matrix has repeated eigenvalues?

**Tip:** Always ensure eigenvectors are normalized when using them in applications, as their direction is often more important than their magnitude.

Learn how to find eigenvalues and eigenvectors of the matrix A = [1 3 3; 1 4 3; -1 3 4]. This problem covers linear algebra concepts, including the characteristic equation, eigenvalue calculations, and eigenvector calculations. Suitable for college-level students and anyone interested in linear algebra.

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