https://github.com/eigenvivek/gershgorin

Utility package for visualizing the Gerhsgorin discs of a matrix.
https://github.com/eigenvivek/gershgorin

Last synced: 18 days ago
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Utility package for visualizing the Gerhsgorin discs of a matrix.

• Host: GitHub
• URL: https://github.com/eigenvivek/gershgorin
• Owner: eigenvivek
• License: mit
• Created: 2022-03-07T18:37:42.000Z (over 2 years ago)
• Default Branch: main
• Last Pushed: 2022-03-12T16:40:02.000Z (over 2 years ago)
• Last Synced: 2024-10-29T07:39:10.955Z (22 days ago)
• Language: Python
• Size: 11.7 KB
• Stars: 0
• Watchers: 1
• Forks: 0
• Open Issues: 0
• Metadata Files:
  • Readme: README.md
  • License: LICENSE

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README

        
# Gershgorin

Visualize the Gershgorin discs that bound the spectrum of a square matrix (see the [Gershgorin disc theorem](https://en.wikipedia.org/wiki/Gershgorin_circle_theorem)).

## Installation

```

python -m pip install gershgorin

```

## Usage

Visualize the Gershgorin discs for a random [Markov matrix](https://en.wikipedia.org/wiki/Stochastic_matrix) shows us that its eigenvalues are bounded within the complex unit circle, guaranteeing convergence of the Markov chain.

```python

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()

```

![markov](https://user-images.githubusercontent.com/29757116/157113695-117246b1-5f82-46db-90a7-536850769a4d.png)

## Useful Tips

- 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](https://www.wikiwand.com/en/Gershgorin_circle_theorem#Strengthening_of_the_theorem) that can be used to strengthen the interpretation of the discs of a matrix