Math Problem Statement
Find the singular value decomposition of the above
Solution
from PIL import Image import pytesseract
Load the image
image_path = '/mnt/data/1732496046501rlph9bgw.jpg' image = Image.open(image_path)
Perform OCR to extract text
extracted_text = pytesseract.image_to_string(image) extracted_text'Find the singular value decomposition of A =\n\nPee\n\nWRN\n''Find the singular value decomposition of A =\n\nPee\n\nWRN\n'successIt seems the OCR extraction was not perfect. However, the question and matrix are clear from the provided image:
We are to find the Singular Value Decomposition (SVD) of the matrix:
\begin{pmatrix} 1 & 2 \\ 1 & 1 \\ 1 & 3 \end{pmatrix}$$ Let me compute the SVD for this matrix.import numpy as np from numpy.linalg import svd # Define the matrix A A = np.array([[1, 2], [1, 1], [1, 3]]) # Perform Singular Value Decomposition U, Sigma, VT = svd(A) # Display the results U, Sigma, VT(array([[-0.54750702, -0.18321952, -0.81649658], [-0.32310986, -0.85377594, 0.40824829], [-0.77190419, 0.4873369 , 0.40824829]]), array([4.07914333, 0.60049122]), array([[-0.40266324, -0.91534819], [-0.91534819, 0.40266324]]))(array([[-0.54750702, -0.18321952, -0.81649658], [-0.32310986, -0.85377594, 0.40824829], [-0.77190419, 0.4873369 , 0.40824829]]), array([4.07914333, 0.60049122]), array([[-0.40266324, -0.91534819], [-0.91534819, 0.40266324]]))successThe Singular Value Decomposition (SVD) of the matrix $$A$$ is as follows: 1. Matrix $$U$$: $$U = \begin{pmatrix} -0.5475 & -0.1832 & -0.8165 \\ -0.3231 & -0.8538 & 0.4082 \\ -0.7719 & 0.4873 & 0.4082 \end{pmatrix}$$ 2. Singular values (diagonal entries of $$\Sigma$$): $$\Sigma = \begin{pmatrix} 4.0791 & 0 \\ 0 & 0.6005 \end{pmatrix}$$ 3. Matrix $$V^T$$ (transpose of $$V$$): $$V^T = \begin{pmatrix} -0.4027 & -0.9153 \\ -0.9153 & 0.4027 \end{pmatrix}$$ This decomposes $$A$$ as $$A = U \Sigma V^T$$. Let me know if you'd like me to verify this or compute any specific part in more detail! ### Related Questions: 1. How does the SVD decomposition help in solving systems of equations? 2. What are the properties of the matrices $$U$$, $$\Sigma$$, and $$V$$ in SVD? 3. Can we use SVD to compute the rank of the matrix $$A$$? 4. How does SVD relate to Principal Component Analysis (PCA)? 5. What is the geometric interpretation of the singular values in $$\Sigma$$? ### Tip: In SVD, the singular values in $$\Sigma$$ provide insight into the condition number of the matrix, which determines how sensitive a matrix is to numerical errors.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Matrix Decomposition
Singular Value Decomposition (SVD)
Formulas
A = UΣVᵀ (Singular Value Decomposition formula)
Theorems
Spectral Theorem
Properties of Orthogonal Matrices
Suitable Grade Level
Undergraduate (Math/Engineering)