A singular value decomposition of A is as follows: 0.5 0.5 0.5 -0.5 -0.5 0.5 0.5 -0.5 -0.5 -0.5 0.5 0.5] [5 01 0.5 0.5 -0.5 0.5 Find the least-squares solution of the linear system -1 A=UEVT Ax = b, where b â₁ Ĵ₂ = - 3 2 4 0 5 [0.6 0 0 0.8 0 0 -0.8] 0.6
A singular value decomposition of A is as follows: 0.5 0.5 0.5 -0.5 -0.5 0.5 0.5 -0.5 -0.5 -0.5 0.5 0.5] [5 01 0.5 0.5 -0.5 0.5 Find the least-squares solution of the linear system -1 A=UEVT Ax = b, where b â₁ Ĵ₂ = - 3 2 4 0 5 [0.6 0 0 0.8 0 0 -0.8] 0.6
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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![**Least-Squares Solution Using Singular Value Decomposition (SVD)**
**Singular Value Decomposition:**
A singular value decomposition (SVD) of \( A \) is given as follows:
\[
A = U \Sigma V^T =
\begin{bmatrix}
0.5 & 0.5 & 0.5 & 0.5 \\
-0.5 & -0.5 & 0.5 & 0.5 \\
0.5 & -0.5 & 0.5 & -0.5 \\
-0.5 & 0.5 & -0.5 & 0.5
\end{bmatrix}
\begin{bmatrix}
5 & 0 \\
0 & 5 \\
0 & 0 \\
0 & 0
\end{bmatrix}
\begin{bmatrix}
0.6 & -0.8 \\
0.8 & 0.6
\end{bmatrix}
\]
**Objective:**
Find the least-squares solution of the linear system:
\[ Ax = b, \text{ where } b =
\begin{bmatrix}
-1 \\
3 \\
2 \\
4
\end{bmatrix}
\]
**Solutions:**
\[
\hat{x}_1 = \; \_\_\_;
\]
\[
\hat{x}_2 = \; \_\_\_;
\]
By solving the system using the given SVD of the matrix \( A \), we can find the least-squares solution \( \hat{x} \).
**Explanation of Graphs/Diagrams:**
Here we have a matrix \( A \) decomposed into three matrices \( U \), \( \Sigma \), and \( V^T \).
- \( U \) is a 4x4 orthogonal matrix.
- \( \Sigma \) is a 4x2 diagonal matrix with non-negative real numbers on the diagonal (the singular values).
- \( V^T \) is the transpose of a 2x2 orthogonal matrix.
Using this decomposition, we find \( x \) that minimizes the equation \( Ax = b \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F80624252-1099-44a7-a129-6e4dc1a5c385%2F5957caf0-34f7-49d4-9c61-11b77368bb86%2Fmizpipi_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Least-Squares Solution Using Singular Value Decomposition (SVD)**
**Singular Value Decomposition:**
A singular value decomposition (SVD) of \( A \) is given as follows:
\[
A = U \Sigma V^T =
\begin{bmatrix}
0.5 & 0.5 & 0.5 & 0.5 \\
-0.5 & -0.5 & 0.5 & 0.5 \\
0.5 & -0.5 & 0.5 & -0.5 \\
-0.5 & 0.5 & -0.5 & 0.5
\end{bmatrix}
\begin{bmatrix}
5 & 0 \\
0 & 5 \\
0 & 0 \\
0 & 0
\end{bmatrix}
\begin{bmatrix}
0.6 & -0.8 \\
0.8 & 0.6
\end{bmatrix}
\]
**Objective:**
Find the least-squares solution of the linear system:
\[ Ax = b, \text{ where } b =
\begin{bmatrix}
-1 \\
3 \\
2 \\
4
\end{bmatrix}
\]
**Solutions:**
\[
\hat{x}_1 = \; \_\_\_;
\]
\[
\hat{x}_2 = \; \_\_\_;
\]
By solving the system using the given SVD of the matrix \( A \), we can find the least-squares solution \( \hat{x} \).
**Explanation of Graphs/Diagrams:**
Here we have a matrix \( A \) decomposed into three matrices \( U \), \( \Sigma \), and \( V^T \).
- \( U \) is a 4x4 orthogonal matrix.
- \( \Sigma \) is a 4x2 diagonal matrix with non-negative real numbers on the diagonal (the singular values).
- \( V^T \) is the transpose of a 2x2 orthogonal matrix.
Using this decomposition, we find \( x \) that minimizes the equation \( Ax = b \).
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