Visualize the Gershgorin discs that bound the spectrum of a square matrix (see the Gershgorin disc theorem).

python -m pip install gershgorin

Visualize the Gershgorin discs for a random Markov matrix shows us that its eigenvalues are bounded within the complex unit circle, guaranteeing convergence of the Markov chain.

import numpy as np
import matplotlib.pyplot as plt
from gershgorin import gershgorin

np.random.seed()

# Sample a random Markov matrix
n = 4
A = np.random.randint(0, 10, (n, n))
A = A / np.sum(A, axis=0)

# Plot the Gershgorin discs
ax = gershgorin(A, annotate=True)

# Add the complex unit circle
x = np.linspace(0, 2 * np.pi, 200)
ax.plot(np.cos(x), np.sin(x), "k--", linewidth=1)

plt.show()

• A matrix and its transpose have the same eigenvalues, so the intersection of the Gershgorin discs of a matrix and its transpose will bound the eigenvalues (can be tighter than either individual region)
• If a matrix is symmetric, its eigenvalues are guaranteed to be real-valued (by the Spectral Theorem), so you can restrict your Gershgorin region to the intersection of the discs and the real axis
• There are many corollaries of Gershgorin's Theorem that can be used to strengthen the interpretation of the discs of a matrix