**Question** The matrix \[ A = \begin{bmatrix} 2 & 5 & 4 \\ 6 & 3 & 0 \\ 6 & 3 & 0 \\ 2 & 5 & 4 \end{bmatrix} \] has singular value decomposition. \[ \begin{bmatrix} \frac{1}{2} & \frac{1}{2} & \frac{1}{2} & \frac{1}{2} \\ \frac{1}{2} & -\frac{1}{2} & -\frac{1}{2} & \frac{1}{2} \\ -\frac{1}{2} & \frac{1}{2} & -\frac{1}{2} & -\frac{1}{2} \\ \frac{1}{2} & \frac{1}{2} & -\frac{1}{2} & -\frac{1}{2} \end{bmatrix} \] \[ \begin{bmatrix} 12 & 0 & 0 \\ 0 & 6 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix} \] \[ \begin{bmatrix} \frac{2}{3} & \frac{2}{3} & \frac{1}{3} \\ -\frac{2}{3} & \frac{1}{3} & \frac{2}{3} \\ \frac{1}{3} & -\frac{2}{3} & \frac{2}{3} \end{bmatrix} \] (a) Use the singular value decomposition to find orthonormal bases for \( R(A^T) \) and \( N(A) \). (b) Use the singular value decomposition to find orthonormal bases for \( R(A) \) and \( N(A^T) \).
**Question** The matrix \[ A = \begin{bmatrix} 2 & 5 & 4 \\ 6 & 3 & 0 \\ 6 & 3 & 0 \\ 2 & 5 & 4 \end{bmatrix} \] has singular value decomposition. \[ \begin{bmatrix} \frac{1}{2} & \frac{1}{2} & \frac{1}{2} & \frac{1}{2} \\ \frac{1}{2} & -\frac{1}{2} & -\frac{1}{2} & \frac{1}{2} \\ -\frac{1}{2} & \frac{1}{2} & -\frac{1}{2} & -\frac{1}{2} \\ \frac{1}{2} & \frac{1}{2} & -\frac{1}{2} & -\frac{1}{2} \end{bmatrix} \] \[ \begin{bmatrix} 12 & 0 & 0 \\ 0 & 6 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix} \] \[ \begin{bmatrix} \frac{2}{3} & \frac{2}{3} & \frac{1}{3} \\ -\frac{2}{3} & \frac{1}{3} & \frac{2}{3} \\ \frac{1}{3} & -\frac{2}{3} & \frac{2}{3} \end{bmatrix} \] (a) Use the singular value decomposition to find orthonormal bases for \( R(A^T) \) and \( N(A) \). (b) Use the singular value decomposition to find orthonormal bases for \( R(A) \) and \( N(A^T) \).
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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![**Question**
The matrix
\[ A = \begin{bmatrix} 2 & 5 & 4 \\ 6 & 3 & 0 \\ 6 & 3 & 0 \\ 2 & 5 & 4 \end{bmatrix} \]
has singular value decomposition.
\[
\begin{bmatrix}
\frac{1}{2} & \frac{1}{2} & \frac{1}{2} & \frac{1}{2} \\
\frac{1}{2} & -\frac{1}{2} & -\frac{1}{2} & \frac{1}{2} \\
-\frac{1}{2} & \frac{1}{2} & -\frac{1}{2} & -\frac{1}{2} \\
\frac{1}{2} & \frac{1}{2} & -\frac{1}{2} & -\frac{1}{2}
\end{bmatrix}
\]
\[
\begin{bmatrix}
12 & 0 & 0 \\
0 & 6 & 0 \\
0 & 0 & 0 \\
0 & 0 & 0
\end{bmatrix}
\]
\[
\begin{bmatrix}
\frac{2}{3} & \frac{2}{3} & \frac{1}{3} \\
-\frac{2}{3} & \frac{1}{3} & \frac{2}{3} \\
\frac{1}{3} & -\frac{2}{3} & \frac{2}{3}
\end{bmatrix}
\]
(a) Use the singular value decomposition to find orthonormal bases for \( R(A^T) \) and \( N(A) \).
(b) Use the singular value decomposition to find orthonormal bases for \( R(A) \) and \( N(A^T) \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3fdbb1a7-d573-479c-8d2a-d1c6b6fb9ae3%2F0e12122a-84d5-4251-bbf6-5896bae67e17%2Fg305qiq_processed.png&w=3840&q=75)
Transcribed Image Text:**Question**
The matrix
\[ A = \begin{bmatrix} 2 & 5 & 4 \\ 6 & 3 & 0 \\ 6 & 3 & 0 \\ 2 & 5 & 4 \end{bmatrix} \]
has singular value decomposition.
\[
\begin{bmatrix}
\frac{1}{2} & \frac{1}{2} & \frac{1}{2} & \frac{1}{2} \\
\frac{1}{2} & -\frac{1}{2} & -\frac{1}{2} & \frac{1}{2} \\
-\frac{1}{2} & \frac{1}{2} & -\frac{1}{2} & -\frac{1}{2} \\
\frac{1}{2} & \frac{1}{2} & -\frac{1}{2} & -\frac{1}{2}
\end{bmatrix}
\]
\[
\begin{bmatrix}
12 & 0 & 0 \\
0 & 6 & 0 \\
0 & 0 & 0 \\
0 & 0 & 0
\end{bmatrix}
\]
\[
\begin{bmatrix}
\frac{2}{3} & \frac{2}{3} & \frac{1}{3} \\
-\frac{2}{3} & \frac{1}{3} & \frac{2}{3} \\
\frac{1}{3} & -\frac{2}{3} & \frac{2}{3}
\end{bmatrix}
\]
(a) Use the singular value decomposition to find orthonormal bases for \( R(A^T) \) and \( N(A) \).
(b) Use the singular value decomposition to find orthonormal bases for \( R(A) \) and \( N(A^T) \).
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