Question:

Keeping the order of diagonal elements when diagonalizing a matrix

Solution:

To compute the eigenvalues of a matrix If you are using numpy.linalg.eig, this means the eigenvalues returned are not necessarily ordered in any particular way. Go through the >numpy manual as it will help you write the code better.

To address this, we can take a sample matrix in which the small but non-zero off-diagonal elements are present, and we can attempt to map them back to their original order.

example_matrix = np.array([[100, 0.2, 0.3], [0.2, 200, 0.5], [0.3, 0.5, 300]])

# Compute eigenvalues and eigenvectors

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

# Original diagonal elements

original_diagonal = np.diag(example_matrix)

# Sort eigenvalues by how close they are to the original diagonal elements

sorted_indices = np.argsort(np.abs(eigenvalues - original_diagonal))

# Sorted eigenvalues and eigenvectors

sorted_eigenvalues = eigenvalues[sorted_indices]

sorted_eigenvectors = eigenvectors[:, sorted_indices]

# Output the sorted eigenvalues and their corresponding eigenvectors

sorted_eigenvalues, sorted_eigenvectors

Result

(array([ 99.99915299, 199.99789408, 300.00295293]),

array([[-0.9999969 , -0.001985 , 0.00150496],

[ 0.0019925 , -0.9999855 , 0.00500279],

[ 0.00149501, 0.00500578, 0.99998635]]))

The eigenvalues and corresponding eigenvectors must be arranged according to how close they are to the original diagonal. To map the eigenvalues back to their original order, you can track the permutation that sorts them in this manner.

>Creating custom required rule - Laravel validation

>How to configure transfers for different accounts in stripe with laravel?

>Laravel Query Solved: Laravel withOnly not restricting the query

>Laravel installation problem: Fix failed to download laravel/laravel from dist- Laravel (v10.2.6)

>How to fix when Postgres connection refused while running laravel tests on gitlab CI?

>Changing main page as Identity's Login page in ASP.NET Core 7

>How can I get the original type of a Mock after the Mock has been cast to an object