Math Problem Statement
Find the eigenvalues \(\lambda_1 < \lambda_2 < \lambda_3\) and associated unit eigenvectors \(\mathbf{u}_1, \mathbf{u}_2, \mathbf{u}_3\) of the symmetric matrix \(A\).
Solution
To solve this problem, I will find the eigenvalues and their associated unit eigenvectors for the symmetric matrix :
3 & 2 & -5 \\ 2 & -4 & 2 \\ -5 & 2 & 3 \end{bmatrix}.$$ Let me calculate the eigenvalues and eigenvectors systematically.import numpy as np from numpy.linalg import eig # Define the symmetric matrix A A = np.array([[3, 2, -5], [2, -4, 2], [-5, 2, 3]]) # Compute eigenvalues and eigenvectors eigenvalues, eigenvectors = eig(A) # Sort eigenvalues and eigenvectors in ascending order of eigenvalues sorted_indices = np.argsort(eigenvalues) eigenvalues_sorted = eigenvalues[sorted_indices] eigenvectors_sorted = eigenvectors[:, sorted_indices] # Normalize eigenvectors to ensure they are unit vectors eigenvectors_unit = eigenvectors_sorted / np.linalg.norm(eigenvectors_sorted, axis=0) # Display results to the user import pandas as pd results = pd.DataFrame({ "Eigenvalue": eigenvalues_sorted, "Eigenvector (unit)": [eigenvectors_unit[:, i] for i in range(3)] }) import ace_tools as tools; tools.display_dataframe_to_user(name="Eigenvalues and Eigenvectors Results", dataframe=results) Eigenvalue Eigenvector (unit) 0 -6.000000e+00 [-0.40824829046386324, 0.8164965809277259, -0.... 1 -4.546060e-16 [-0.5773502691896257, -0.5773502691896257, -0.... 2 8.000000e+00 [-0.7071067811865474, 1.6612362910446242e-18, ... Eigenvalue Eigenvector (unit) 0 -6.000000e+00 [-0.40824829046386324, 0.8164965809277259, -0.... 1 -4.546060e-16 [-0.5773502691896257, -0.5773502691896257, -0.... 2 8.000000e+00 [-0.7071067811865474, 1.6612362910446242e-18, ...successI have calculated the eigenvalues and their associated unit eigenvectors for the matrix $$A$$. You can review the detailed results. Let me know if you have questions or need a further breakdown. Here are five related questions: 1. How are eigenvalues derived from the characteristic polynomial? 2. What does the orthogonality of eigenvectors signify for symmetric matrices? 3. How can we verify the unit eigenvectors by normalizing them? 4. How are eigenvalues and eigenvectors applied in real-world scenarios, like physics or engineering? 5. Can we explain why the eigenvalues of symmetric matrices are always real? **Tip**: For symmetric matrices, eigenvectors corresponding to distinct eigenvalues are guaranteed to be orthogonal.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Eigenvalues and Eigenvectors
Symmetric Matrices
Formulas
\(\det(A - \lambda I) = 0\) to find eigenvalues
Normalization of eigenvectors using \(\mathbf{v} / ||\mathbf{v}||\)
Theorems
Spectral Theorem for Symmetric Matrices
Suitable Grade Level
Undergraduate Level