https://file.aiquickdraw.com/mathbot/file/17378785285657nqodqym.jpg

To diagonalize the matrix \( A \), we will follow these steps:

Given matrix \( A \):
\[
A = \begin{bmatrix}
8 & -8 & -2 \\
4 & -3 & -2 \\
3 & -4 & 1
\end{bmatrix}
\]

### Steps for Diagonalization:

1. **Find the Eigenvalues**: We first need to solve the characteristic equation for eigenvalues. The characteristic equation is:
   \[
   \det(A - \lambda I) = 0
   \]
   Where \( I \) is the identity matrix, and \( \lambda \) represents the eigenvalue.

2. **Find the Eigenvectors**: Once we have the eigenvalues, we substitute each eigenvalue \( \lambda \) into the equation \( (A - \lambda I) v = 0 \) to find the corresponding eigenvectors.

3. **Form the Matrix P**: The matrix \( P \) is made up of the eigenvectors as columns.

4. **Diagonal Matrix D**: The diagonal matrix \( D \) contains the eigenvalues on its diagonal.

The matrix \( A \) will be diagonalized if it is diagonalizable, i.e., if the matrix has a full set of linearly independent eigenvectors.

Let me start by calculating the eigenvalues for this matrix. numpy as np

# Define the matrix A
A = np.array([[8, -8, -2],
              [4, -3, -2],
              [3, -4, 1]])

# Calculate eigenvalues and eigenvectors
eigenvalues, eigenvectors = np.linalg.eig(A)

eigenvalues, eigenvectorsThe eigenvalues of the matrix \( A \) are:

\[
\lambda_1 = 1, \quad \lambda_2 = 3, \quad \lambda_3 = 2
\]

The corresponding eigenvectors are:

For \( \lambda_1 = 1 \):
\[
v_1 = \begin{bmatrix} -0.7428 \\ -0.5571 \\ -0.3714 \end{bmatrix}
\]

For \( \lambda_2 = 3 \):
\[
v_2 = \begin{bmatrix} -0.8165 \\ -0.4082 \\ -0.4082 \end{bmatrix}
\]

For \( \lambda_3 = 2 \):
\[
v_3 = \begin{bmatrix} -0.8018 \\ -0.5345 \\ -0.2673 \end{bmatrix}
\]

### Forming the Matrix \( P \):

Now we can form the matrix \( P \) by using the eigenvectors as columns:

\[
P = \begin{bmatrix} v_1 & v_2 & v_3 \end{bmatrix}
\]

The diagonal matrix \( D \) will be:

\[
D = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 2 \end{bmatrix}
\]

Would you like further details on the steps or explanation?

Diagonalize the matrix A = [[8, -8, -2], [4, -3, -2], [3, -4, 1]].

Explore the step-by-step diagonalization of matrix A = [[8, -8, -2], [4, -3, -2], [3, -4, 1]]. Learn about eigenvalues, eigenvectors, and how to construct the diagonal and transformation matrices. Suitable for undergraduate linear algebra students.

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