To find the invertible matrix $P$ such that $D = P^{-1}AP$ is a diagonal matrix, follow these steps:

Step 1: Find the eigenvalues of $A$

The eigenvalues of $A$ are the roots of the characteristic equation: $\det(A - \lambda I) = 0$ Here, $I$ is the identity matrix and $\lambda$ represents the eigenvalues.

Step 2: Find the eigenvectors of $A$

For each eigenvalue $\lambda$, solve $(A - \lambda I)x = 0$ to find the corresponding eigenvector(s). These eigenvectors will form the columns of the matrix $P$.

Step 3: Construct $P$ and $D$

• $P$: Matrix whose columns are the eigenvectors of $A$.
• $D$: Diagonal matrix whose diagonal entries are the eigenvalues of $A$.

Let's compute these step by step.import numpy as np

Define the matrix A

A = np.array([ [-3, 2, -8], [4, -1, 8], [2, -2, 7] ])

Step 1: Find eigenvalues and eigenvectors

eigenvalues, eigenvectors = np.linalg.eig(A)

Step 2: Construct the diagonal matrix D and matrix P

D = np.diag(eigenvalues) P = eigenvectors

eigenvalues, P, D(array([-1., 1., 3.]), array([[ 0.66666667, 0.40824829, -0.57735027], [-0.66666667, -0.81649658, 0.57735027], [-0.33333333, -0.40824829, 0.57735027]]), array([[-1., 0., 0.], [ 0., 1., 0.], [ 0., 0., 3.]]))(array([-1., 1., 3.]), array([[ 0.66666667, 0.40824829, -0.57735027], [-0.66666667, -0.81649658, 0.57735027], [-0.33333333, -0.40824829, 0.57735027]]), array([[-1., 0., 0.], [ 0., 1., 0.], [ 0., 0., 3.]]))successThe eigenvalues and eigenvectors of $A$ are as follows:

Eigenvalues

$\lambda_1 = -1, \quad \lambda_2 = 1, \quad \lambda_3 = 3$

Matrix $P$ (formed by eigenvectors as columns)