1 0 12
2 −5 0
1 0 2      Find the Eigen values and Eigen vectors of the following matrices

To find the eigenvalues and eigenvectors of a given matrix, we can use the following procedure:

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

### Step 2: Find Eigenvectors
For each eigenvalue \(\lambda\), the corresponding eigenvector is found by solving the system of equations:
\[
(A - \lambda I)v = 0
\]
Where \(v\) is the eigenvector corresponding to eigenvalue \(\lambda\).

Let’s now solve this step-by-step for your matrix:

\[
A = \begin{pmatrix}
1 & 0 & 12 \\
2 & -5 & 0 \\
1 & 0 & 2
\end{pmatrix}
\]

### Step 1: Eigenvalues

We need to calculate the determinant of \( A - \lambda I \), which is:
\[
A - \lambda I = \begin{pmatrix}
1 - \lambda & 0 & 12 \\
2 & -5 - \lambda & 0 \\
1 & 0 & 2 - \lambda
\end{pmatrix}
\]

The characteristic equation is:
\[
\det(A - \lambda I) = 0
\]
Let’s compute this determinant.The eigenvalues of the matrix \( A \) are:

\[
\lambda_1 = -5, \quad \lambda_2 = -2, \quad \lambda_3 = 5
\]

The corresponding eigenvectors are:

\[
v_1 = \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix}, \quad
v_2 = \begin{pmatrix} -0.815 \\ -0.543 \\ 0.204 \end{pmatrix}, \quad
v_3 = \begin{pmatrix} -0.932 \\ -0.186 \\ -0.311 \end{pmatrix}
\]

Would you like more details on how these were computed?

### Related Questions:
1. How can eigenvalues and eigenvectors be used in linear transformations?
2. What is the geometric interpretation of eigenvectors?
3. How do you verify if a given vector is an eigenvector for a matrix?
4. Can eigenvalues ever be complex for real matrices?
5. How does diagonalization relate to eigenvalues?

### Tip:
Eigenvalues represent scaling factors, while eigenvectors indicate the direction of the transformation.

1 0 12
2 −5 0
1 0 2 Find the Eigen values and Eigen vectors of the following matrices

This page explains how to find the eigenvalues and eigenvectors of the matrix [1 0 12; 2 −5 0; 1 0 2] using the characteristic equation and determinant. Learn key concepts in linear algebra and matrix theory, suitable for undergraduate students.

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