1. Given that the matrix A has the Singular Value Decomposition A = UΣ V, where: 1 1 6 6 000 2/2 3. U = 0 0 V = V2 0 0 V2 3 6 (a) Write down an orthonormal basis for R(A). (b) Write down an orthonormal basis for N(A). (c) Write down an orthonormal basis for R(AT). (d) Write down an orthonormal basis for N(AT).
1. Given that the matrix A has the Singular Value Decomposition A = UΣ V, where: 1 1 6 6 000 2/2 3. U = 0 0 V = V2 0 0 V2 3 6 (a) Write down an orthonormal basis for R(A). (b) Write down an orthonormal basis for N(A). (c) Write down an orthonormal basis for R(AT). (d) Write down an orthonormal basis for N(AT).
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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![### Singular Value Decomposition of a Matrix
#### Problem Statement
1. Given that the matrix \( A \) has the Singular Value Decomposition \( A = U \Sigma V^T \), where:
\[ U = \begin{pmatrix}
\frac{2}{3} & \frac{1}{\sqrt{2}} & -\frac{\sqrt{2}}{6} \\
\frac{1}{3} & 0 & \frac{2\sqrt{2}}{3} \\
-\frac{2}{3} & \frac{1}{\sqrt{2}} & \frac{\sqrt{2}}{6}
\end{pmatrix} \]
\[ \Sigma = \begin{pmatrix}
6 & 0 & 0 & 0 \\
0 & 4 & 0 & 0 \\
0 & 0 & 0 & 0
\end{pmatrix} \]
\[ V = \begin{pmatrix}
\frac{1}{2} & 0 & \frac{1}{\sqrt{2}} & \frac{1}{2} \\
-\frac{1}{2} & -\frac{1}{\sqrt{2}} & 0 & \frac{1}{2} \\
\frac{1}{2} & -\frac{1}{\sqrt{2}} & 0 & -\frac{1}{2} \\
-\frac{1}{2} & 0 & \frac{1}{\sqrt{2}} & -\frac{1}{2}
\end{pmatrix} \]
(a) Write down an orthonormal basis for \( R(A) \).
(b) Write down an orthonormal basis for \( N(A) \).
(c) Write down an orthonormal basis for \( R(A^T) \).
(d) Write down an orthonormal basis for \( N(A^T) \).
#### Solution
Given the Singular Value Decomposition \( A = U \Sigma V^T \), we can determine the orthonormal bases for the four fundamental subspaces of \( A \).
(a) **Orthonormal Basis for \( R(A) \)**
The range (column space) of \( A \), \( R(A) \), is spanned by the columns of \( U \) corresponding to the non-zero singular values in \(](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb8d14a18-18b6-4b9f-91b0-8c8acc9b34e2%2F889824b2-5526-4975-b5aa-e14e36d0a932%2Fn8meqab_processed.png&w=3840&q=75)
Transcribed Image Text:### Singular Value Decomposition of a Matrix
#### Problem Statement
1. Given that the matrix \( A \) has the Singular Value Decomposition \( A = U \Sigma V^T \), where:
\[ U = \begin{pmatrix}
\frac{2}{3} & \frac{1}{\sqrt{2}} & -\frac{\sqrt{2}}{6} \\
\frac{1}{3} & 0 & \frac{2\sqrt{2}}{3} \\
-\frac{2}{3} & \frac{1}{\sqrt{2}} & \frac{\sqrt{2}}{6}
\end{pmatrix} \]
\[ \Sigma = \begin{pmatrix}
6 & 0 & 0 & 0 \\
0 & 4 & 0 & 0 \\
0 & 0 & 0 & 0
\end{pmatrix} \]
\[ V = \begin{pmatrix}
\frac{1}{2} & 0 & \frac{1}{\sqrt{2}} & \frac{1}{2} \\
-\frac{1}{2} & -\frac{1}{\sqrt{2}} & 0 & \frac{1}{2} \\
\frac{1}{2} & -\frac{1}{\sqrt{2}} & 0 & -\frac{1}{2} \\
-\frac{1}{2} & 0 & \frac{1}{\sqrt{2}} & -\frac{1}{2}
\end{pmatrix} \]
(a) Write down an orthonormal basis for \( R(A) \).
(b) Write down an orthonormal basis for \( N(A) \).
(c) Write down an orthonormal basis for \( R(A^T) \).
(d) Write down an orthonormal basis for \( N(A^T) \).
#### Solution
Given the Singular Value Decomposition \( A = U \Sigma V^T \), we can determine the orthonormal bases for the four fundamental subspaces of \( A \).
(a) **Orthonormal Basis for \( R(A) \)**
The range (column space) of \( A \), \( R(A) \), is spanned by the columns of \( U \) corresponding to the non-zero singular values in \(
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