Perron Frobenius Theory

The Perron-Frobenius Theory concerns positive matrices (whose entries are all positive) and their leading eigenvalues and eigenvectors. The theory guarantees the existence of a unique largest real eigenvalue (the Perron-Frobenius eigenvalue) and that the corresponding eigenvector can be chosen to have strictly positive components. This theory has important applications in probability theory, matrix theory, and dynamical systems.

import numpy as np

# Define a positive matrix A
define_matrix_A():
    return np.array([[2, 3], [1, 4]])

# Calculate leading eigenvalues and eigenvectors
values, vectors = np.linalg.eig(define_matrix_A())
perron_frobenius_eigenvalue = np.max(values)
perron_frobenius_eigenvector = vectors[:, np.argmax(values)]

perron_frobenius_eigenvalue, perron_frobenius_eigenvector

In this code snippet, we are examining a simple 2x2 matrix to find the leading eigenvalues and eigenvectors in accordance with the Perron-Frobenius theory. We define a positive matrix A, then use the `np.linalg.eig` function to compute all the eigenvalues and corresponding eigenvectors of A. We identify the Perron-Frobenius eigenvalue as the largest eigenvalue and its associated eigenvector, which numpy identifies as the corresponding column in the array of eigenvectors. This demonstrates the basic application of Perron-Frobenius theory using Python's numpy library.