A singular value decomposition of a matrix A is as follows: 0.5 -0.5 -0.5] [5 01 0.5 0.5 0.5 0 5 0.5 0.5 -0.5 0 -0.5 0.5 -0.5 0.5 00 1. Find the closest (with respect to the Frobenius norm) matrix of rank 1 to A. A = Someone A1 = 0.5 0.5 2. Find the Frobenius norm of A - A1. ||A - A1||F janu 0.6 0.87 0.8 0.6
A singular value decomposition of a matrix A is as follows: 0.5 -0.5 -0.5] [5 01 0.5 0.5 0.5 0 5 0.5 0.5 -0.5 0 -0.5 0.5 -0.5 0.5 00 1. Find the closest (with respect to the Frobenius norm) matrix of rank 1 to A. A = Someone A1 = 0.5 0.5 2. Find the Frobenius norm of A - A1. ||A - A1||F janu 0.6 0.87 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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![### Singular Value Decomposition of a Matrix
A singular value decomposition of a matrix \( A \) is as follows:
\[
A = \begin{bmatrix}
0.5 & 0.5 & -0.5 & -0.5 \\
0.5 & 0.5 & 0.5 & 0.5 \\
-0.5 & 0.5 & 0.5 & -0.5 \\
-0.5 & 0.5 & -0.5 & 0.5
\end{bmatrix}
\begin{bmatrix}
5 & 0 & 0 & 0 \\
0 & 5 & 0 & 0 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0
\end{bmatrix}
\begin{bmatrix}
0.6 & 0.8 \\
-0.8 & 0.6
\end{bmatrix}
\]
### Tasks
1. **Find the Closest Rank 1 Approximation of Matrix \( A \)**
Find the closest (with respect to the Frobenius norm) matrix of rank 1 to \( A \):
\[
A1 = \begin{bmatrix}
\boxed{\phantom{0}} & \boxed{\phantom{0}} & \boxed{\phantom{0}} & \boxed{\phantom{0}} \\
\boxed{\phantom{0}} & \boxed{\phantom{0}} & \boxed{\phantom{0}} & \boxed{\phantom{0}} \\
\boxed{\phantom{0}} & \boxed{\phantom{0}} & \boxed{\phantom{0}} & \boxed{\phantom{0}} \\
\boxed{\phantom{0}} & \boxed{\phantom{0}} & \boxed{\phantom{0}} & \boxed{\phantom{0}}
\end{bmatrix}
\]
2. **Find the Frobenius Norm of \( A - A1 \)**
Calculate the Frobenius norm of the difference between \( A \) and \( A1 \):
\[
\| A - A1 \|_{F} = \boxed{\phantom{0}}
\]
###](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F80624252-1099-44a7-a129-6e4dc1a5c385%2F4e8758cc-d421-4dc3-b62f-7e7d116acc40%2F7rrheh5_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:
\[
A = \begin{bmatrix}
0.5 & 0.5 & -0.5 & -0.5 \\
0.5 & 0.5 & 0.5 & 0.5 \\
-0.5 & 0.5 & 0.5 & -0.5 \\
-0.5 & 0.5 & -0.5 & 0.5
\end{bmatrix}
\begin{bmatrix}
5 & 0 & 0 & 0 \\
0 & 5 & 0 & 0 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0
\end{bmatrix}
\begin{bmatrix}
0.6 & 0.8 \\
-0.8 & 0.6
\end{bmatrix}
\]
### Tasks
1. **Find the Closest Rank 1 Approximation of Matrix \( A \)**
Find the closest (with respect to the Frobenius norm) matrix of rank 1 to \( A \):
\[
A1 = \begin{bmatrix}
\boxed{\phantom{0}} & \boxed{\phantom{0}} & \boxed{\phantom{0}} & \boxed{\phantom{0}} \\
\boxed{\phantom{0}} & \boxed{\phantom{0}} & \boxed{\phantom{0}} & \boxed{\phantom{0}} \\
\boxed{\phantom{0}} & \boxed{\phantom{0}} & \boxed{\phantom{0}} & \boxed{\phantom{0}} \\
\boxed{\phantom{0}} & \boxed{\phantom{0}} & \boxed{\phantom{0}} & \boxed{\phantom{0}}
\end{bmatrix}
\]
2. **Find the Frobenius Norm of \( A - A1 \)**
Calculate the Frobenius norm of the difference between \( A \) and \( A1 \):
\[
\| A - A1 \|_{F} = \boxed{\phantom{0}}
\]
###
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