A singular value decomposition of a matrix A is as follows: -0.5 0.5 0.5 -0.5] [10 01 0.5 0 TEJ 0.5 -0.5 0 (a) Find the closest (with respect to the Frobenlus norm) matrix of rank 1 to A. A = A1 = 0.5 0.5 0.5 -0.5 0.5 -0.5 0 -0.5 0.5 0.5 (a) Find the Frobenius norm of A - A1. ||A-A1|| = (c) Find the rank of A rank(A) 0090 5 0.8 0.6 -0.6 0.8
A singular value decomposition of a matrix A is as follows: -0.5 0.5 0.5 -0.5] [10 01 0.5 0 TEJ 0.5 -0.5 0 (a) Find the closest (with respect to the Frobenlus norm) matrix of rank 1 to A. A = A1 = 0.5 0.5 0.5 -0.5 0.5 -0.5 0 -0.5 0.5 0.5 (a) Find the Frobenius norm of A - A1. ||A-A1|| = (c) Find the rank of A rank(A) 0090 5 0.8 0.6 -0.6 0.8
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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![### Singular Value Decomposition of a Matrix
A singular value decomposition of a matrix \( A \) is as follows:
\[
A = \begin{bmatrix}
-0.5 & 0.5 & 0.5 & -0.5 \\
0.5 & 0.5 & 0.5 & 0.5 \\
0.5 & -0.5 & 0.5 & 0.5 \\
-0.5 & -0.5 & 0.5 & 0.5 \\
\end{bmatrix}
\begin{bmatrix}
10 & 0 \\
0 & 5 \\
0 & 0 \\
0 & 0 \\
\end{bmatrix}
\begin{bmatrix}
0.8 & 0.6 \\
-0.6 & 0.8 \\
\end{bmatrix}
\]
#### (a) Closest Rank-1 Approximation
Find the closest (i.e., with respect to the Frobenius norm) matrix of rank 1 to \( A \):
\[
A_1 = \begin{bmatrix}
\_\_ & \_\_ & \_\_ & \_\_ \\
\_\_ & \_\_ & \_\_ & \_\_ \\
\_\_ & \_\_ & \_\_ & \_\_ \\
\_\_ & \_\_ & \_\_ & \_\_ \\
\end{bmatrix}
\]
#### (b) Frobenius Norm
Find the Frobenius norm of \( A - A_1 \):
\[
\|A - A_1\|_F = \_\_
\]
#### (c) Rank of A
Find the rank of \( A \):
\[
\text{rank}(A) = \_\_
\]
---
### Explanation of Diagrams
The diagram provided includes three matrices involved in the singular value decomposition of matrix \( A \):
1. **Matrix \( A \)**: A \( 4 \times 4 \) matrix composed of real numbers.
2. **Diagonal Matrix**: A \( 4 \times 2 \) matrix where only the main diagonal elements are non-zero (here, 10 and 5).
3. **Matrix \( V \)**: A \( 2 \times 2 \) orthogonal](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffb2133c9-e1e5-4d56-9c72-044227328930%2F26b46325-cc58-44ad-9238-7b54dc1daab6%2F8qrx9en_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Singular Value Decomposition of a Matrix
A singular value decomposition of a matrix \( A \) is as follows:
\[
A = \begin{bmatrix}
-0.5 & 0.5 & 0.5 & -0.5 \\
0.5 & 0.5 & 0.5 & 0.5 \\
0.5 & -0.5 & 0.5 & 0.5 \\
-0.5 & -0.5 & 0.5 & 0.5 \\
\end{bmatrix}
\begin{bmatrix}
10 & 0 \\
0 & 5 \\
0 & 0 \\
0 & 0 \\
\end{bmatrix}
\begin{bmatrix}
0.8 & 0.6 \\
-0.6 & 0.8 \\
\end{bmatrix}
\]
#### (a) Closest Rank-1 Approximation
Find the closest (i.e., with respect to the Frobenius norm) matrix of rank 1 to \( A \):
\[
A_1 = \begin{bmatrix}
\_\_ & \_\_ & \_\_ & \_\_ \\
\_\_ & \_\_ & \_\_ & \_\_ \\
\_\_ & \_\_ & \_\_ & \_\_ \\
\_\_ & \_\_ & \_\_ & \_\_ \\
\end{bmatrix}
\]
#### (b) Frobenius Norm
Find the Frobenius norm of \( A - A_1 \):
\[
\|A - A_1\|_F = \_\_
\]
#### (c) Rank of A
Find the rank of \( A \):
\[
\text{rank}(A) = \_\_
\]
---
### Explanation of Diagrams
The diagram provided includes three matrices involved in the singular value decomposition of matrix \( A \):
1. **Matrix \( A \)**: A \( 4 \times 4 \) matrix composed of real numbers.
2. **Diagonal Matrix**: A \( 4 \times 2 \) matrix where only the main diagonal elements are non-zero (here, 10 and 5).
3. **Matrix \( V \)**: A \( 2 \times 2 \) orthogonal
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