1 1 3 1 5 1 3 1 1

Find eigan values and eigan vectors

The sequence you provided appears to represent a matrix. If you intended this sequence as a 3x3 matrix, it could be interpreted as follows:

\[
\mathbf{A} = \begin{pmatrix}
1 & 1 & 3 \\
1 & 5 & 1 \\
3 & 1 & 1
\end{pmatrix}
\]

To find the eigenvalues and eigenvectors, we proceed with the following steps:

### 1. Eigenvalues:
The eigenvalues \(\lambda\) of matrix \(\mathbf{A}\) are found by solving the characteristic equation:

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

Where \(\mathbf{I}\) is the identity matrix of the same size as \(\mathbf{A}\). 

### 2. Eigenvectors:
For each eigenvalue \(\lambda\), the corresponding eigenvector \(\mathbf{v}\) is found by solving the equation:

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

Let's compute the eigenvalues and eigenvectors.The eigenvalues and corresponding eigenvectors of the matrix \(\mathbf{A}\) are as follows:

### Eigenvalues:
\[
\lambda_1 = -2, \quad \lambda_2 = 3, \quad \lambda_3 = 6
\]

### Eigenvectors:
- For \(\lambda_1 = -2\), the eigenvector is:
  \[
  \mathbf{v}_1 = \begin{pmatrix} 0.7071 \\ 0 \\ -0.7071 \end{pmatrix}
  \]

- For \(\lambda_2 = 3\), the eigenvector is:
  \[
  \mathbf{v}_2 = \begin{pmatrix} -0.5774 \\ 0.5774 \\ -0.5774 \end{pmatrix}
  \]

- For \(\lambda_3 = 6\), the eigenvector is:
  \[
  \mathbf{v}_3 = \begin{pmatrix} -0.4082 \\ -0.8165 \\ -0.4082 \end{pmatrix}
  \]

Would you like more details or have any questions?

### Related Questions:
1. How do we verify if a given vector is an eigenvector of a matrix?
2. What is the significance of eigenvalues and eigenvectors in linear algebra?
3. How can we interpret the geometric meaning of eigenvectors?
4. Can eigenvalues be complex for certain matrices?
5. What is the application of eigenvalues and eigenvectors in data science?

### Tip:
Eigenvalues and eigenvectors are fundamental in understanding the behavior of linear transformations, especially in applications like principal component analysis (PCA) in machine learning.

Discover how to find eigenvalues and eigenvectors of a 3x3 matrix with a detailed step-by-step solution. Learn about the characteristic equation, eigenvector equations, and their applications in linear algebra. Suitable for students and professionals studying advanced mathematics.

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