4.8 Find the SVD of the matrix 3 2 2 A = 2 3 -2
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![**4.8 Problem Statement:**
Find the Singular Value Decomposition (SVD) of the matrix
\[
A = \begin{bmatrix} 3 & 2 & 2 \\ 2 & 3 & -2 \end{bmatrix}.
\]
**Explanation:**
The problem asks to perform Singular Value Decomposition on matrix \( A \), which transforms matrix \( A \) into the product of three matrices \( U \), \( \Sigma \), and \( V^T \) where \( U \) and \( V \) are orthogonal matrices, and \( \Sigma \) is a diagonal matrix containing the singular values.
The matrix \( A \) is a \( 2 \times 3 \) matrix given by:
\[
A = \begin{bmatrix} 3 & 2 & 2 \\ 2 & 3 & -2 \end{bmatrix}
\]
Further computational steps involve calculating the SVD components:
1. **Compute \( A^T A \) and \( AA^T \)** for eigenvalue decomposition.
2. **Determine the eigenvalues** to find singular values.
3. **Calculate singular vectors** to form matrices \( U \) and \( V \).
4. **Construct the diagonal matrix \( \Sigma \)** with the singular values.
This process captures the essence of decomposing matrices into simpler, valuable components used for various applications like solving systems, data compression, and more. This example focuses on clarity in foundational matrix operations.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd4ed983c-8942-4ca0-93d1-8e13c157c75d%2Fcb5bd6f7-e933-45d9-8de3-5dbf37538df3%2Fraz0q5m_processed.png&w=3840&q=75)
Transcribed Image Text:**4.8 Problem Statement:**
Find the Singular Value Decomposition (SVD) of the matrix
\[
A = \begin{bmatrix} 3 & 2 & 2 \\ 2 & 3 & -2 \end{bmatrix}.
\]
**Explanation:**
The problem asks to perform Singular Value Decomposition on matrix \( A \), which transforms matrix \( A \) into the product of three matrices \( U \), \( \Sigma \), and \( V^T \) where \( U \) and \( V \) are orthogonal matrices, and \( \Sigma \) is a diagonal matrix containing the singular values.
The matrix \( A \) is a \( 2 \times 3 \) matrix given by:
\[
A = \begin{bmatrix} 3 & 2 & 2 \\ 2 & 3 & -2 \end{bmatrix}
\]
Further computational steps involve calculating the SVD components:
1. **Compute \( A^T A \) and \( AA^T \)** for eigenvalue decomposition.
2. **Determine the eigenvalues** to find singular values.
3. **Calculate singular vectors** to form matrices \( U \) and \( V \).
4. **Construct the diagonal matrix \( \Sigma \)** with the singular values.
This process captures the essence of decomposing matrices into simpler, valuable components used for various applications like solving systems, data compression, and more. This example focuses on clarity in foundational matrix operations.
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