[Term Entry] Python - NumPy Linear-algebra: eig() #6416 #6540
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mamtawardhani
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mamtawardhani
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Apr 14, 2025
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Hey @dancikmad, thank you for contributing to Codecademy Docs, the entry is nicely written! 😄
I've suggested a few changes, could you please review and modify those at your earliest convenience? Thank you! 😃
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| **Parameters:** | ||
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| - `a`: array_like of shape (M, M). A square matrix whose eigenvalues and eigenvectors will be computed. |
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| - `a`: array_like of shape (M, M). A square matrix whose eigenvalues and eigenvectors will be computed. | |
| - `a`: A square matrix of shape (M, M) used as input to compute its eigenvalues and corresponding right eigenvectors. |
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| - `w`: ndarray of shape (M,). The eigenvalues of the matrix. | ||
| - `v`: ndarray of shape (M, M). The eigenvectors of the matrix. The column `v[:, i]` corresponds to the eigenvalue `w[i]`. |
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| - `w`: ndarray of shape (M,). The eigenvalues of the matrix. | |
| - `v`: ndarray of shape (M, M). The eigenvectors of the matrix. The column `v[:, i]` corresponds to the eigenvalue `w[i]`. | |
| - `w`: An ndarray of shape (M,), containing the eigenvalues of the matrix. | |
| - `v`: An ndarray of shape (M, M), containing the eigenvectors of the matrix, where each column `v[:, i]` corresponds to the eigenvalue `w[i]`. |
| Are they equal? True | ||
| ``` | ||
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| This example demonstrates how the `eig()` function handles matrices with complex eigenvalues. The rotation matrix B has eigenvalues 1j and -1j, illustrating how complex eigenvalues often indicate rotational transformations. |
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| This example demonstrates how the `eig()` function handles matrices with complex eigenvalues. The rotation matrix B has eigenvalues 1j and -1j, illustrating how complex eigenvalues often indicate rotational transformations. | |
| This example demonstrates how the `eig()` function handles matrices with complex eigenvalues. The rotation matrix B has eigenvalues `1j` and `-1j`, illustrating how complex eigenvalues often indicate rotational transformations. |
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| ## Codebyte Example: Using Eigendecomposition for System Analysis | ||
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| This example uses eigenvalues to analyze a dynamical system: | ||
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| ```codebyte/python | ||
| import numpy as np | ||
| import matplotlib.pyplot as plt | ||
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| # Define a system matrix (represents a linear dynamical system) | ||
| A = np.array([[0.5, 0.1], | ||
| [0.2, 0.8]]) | ||
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| # Compute eigenvalues and eigenvectors | ||
| eigenvalues, eigenvectors = np.linalg.eig(A) | ||
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| print("System matrix:") | ||
| print(A) | ||
| print("\nEigenvalues:", eigenvalues) | ||
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| # Check system stability (all eigenvalues must have magnitude < 1) | ||
| stable = np.all(np.abs(eigenvalues) < 1) | ||
| print("\nIs the system stable?", stable) | ||
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| # Simulate the system evolution | ||
| x0 = np.array([1, 0]) # Initial state | ||
| steps = 20 | ||
| states = [x0] | ||
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| for i in range(steps): | ||
| x0 = A @ x0 | ||
| states.append(x0) | ||
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| states = np.array(states) | ||
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| # Plot the system trajectory | ||
| plt.figure(figsize=(10, 5)) | ||
| plt.plot(range(steps+1), states[:, 0], 'b-', label='x1') | ||
| plt.plot(range(steps+1), states[:, 1], 'r-', label='x2') | ||
| plt.xlabel('Time Step') | ||
| plt.ylabel('State Value') | ||
| plt.title('System Evolution') | ||
| plt.legend() | ||
| plt.grid(True) | ||
| plt.tight_layout() | ||
| plt.show() | ||
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| print("\nFinal state after", steps, "steps:", states[-1]) | ||
| ``` |
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this section can be removed as we have 3 examples above
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| print("Are they equal?", np.allclose(np.dot(A, eigenvectors[:, 0]), | ||
| eigenvalues[0] * eigenvectors[:, 0])) |
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| print("Are they equal?", np.allclose(np.dot(A, eigenvectors[:, 0]), | |
| eigenvalues[0] * eigenvectors[:, 0])) | |
| print("Are they equal?", np.allclose(np.dot(A, eigenvectors[:, 0]), eigenvalues[0] * eigenvectors[:, 0])) |
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| print("Are they equal?", np.allclose(np.dot(B, eigenvectors[:, 0]), | ||
| eigenvalues[0] * eigenvectors[:, 0])) |
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| print("Are they equal?", np.allclose(np.dot(B, eigenvectors[:, 0]), | |
| eigenvalues[0] * eigenvectors[:, 0])) | |
| print("Are they equal?", np.allclose(np.dot(B, eigenvectors[:, 0]), eigenvalues[0] * eigenvectors[:, 0])) |
| Are they equal? True | ||
| ``` | ||
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| This example demonstrates how the `eig()` function handles matrices with complex eigenvalues. The rotation matrix B has eigenvalues 1j and -1j, illustrating how complex eigenvalues often indicate rotational transformations. |
There was a problem hiding this comment.
Suggested change
| This example demonstrates how the `eig()` function handles matrices with complex eigenvalues. The rotation matrix B has eigenvalues 1j and -1j, illustrating how complex eigenvalues often indicate rotational transformations. | |
| This example demonstrates how the `eig()` function handles matrices with complex eigenvalues. The rotation matrix B has eigenvalues `1j` and `-1j`, illustrating how complex eigenvalues often indicate rotational transformations. |
mamtawardhani
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@mamtawardhani thank you for your feedback! I will work on this changes today |
* Make changes after first PR review * Delete codebyte/example - 3 examples already provided * refactor print statements in ```shell:tag``` codes
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@mamtawardhani hi :) i trust you are doing well |
mamtawardhani
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Apr 19, 2025
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Sriparno08
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Apr 25, 2025
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👋 @dancikmad 🎉 Your contribution(s) can be seen here: https://www.codecademy.com/resources/docs/numpy/linear-algebra/eig Please note it may take a little while for changes to become visible. |
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