**Singular Value Decomposition (SVD) of a Matrix**Given that the matrix $ A $ has the Singular Value Decomposition $ A = U \Sigma V^T $, where:\[ U = \begin{pmatrix}\frac{2}{3} & \frac{1}{\sqrt{2}} & -\frac{\sqrt{2}}{6} \\\frac{1}{3} & 0 & \frac{2\sqrt{2}}{3} \\-\frac{2}{3} & \frac{1}{\sqrt{2}} & \frac{\sqrt{2}}{6}\end{pmatrix}, \]\[ \Sigma = \begin{pmatrix}6 & 0 & 0 & 0 \\0 & 4 & 0 & 0 \\0 & 0 & 0 & 0\end{pmatrix}, \]\[ V = \begin{pmatrix}\frac{1}{2} & 0 & \frac{1}{\sqrt{2}} & \frac{1}{2} \\-\frac{1}{2} & -\frac{1}{\sqrt{2}} & 0 & \frac{1}{2} \\\frac{1}{2} & 0 & -\frac{1}{\sqrt{2}} & \frac{1}{2} \\-\frac{1}{2} & \frac{1}{\sqrt{2}} & 0 & \frac{1}{2}\end{pmatrix} \](a) **What is the rank $ r $ of $ A $?**(b) **Write down the compact form of the SVD of $ A $**(c) **Find the rank 1 matrix $ A' $ that is closest to $ A $ (with respect to the Frobenius norm).**

The given matrix: Where, To find A: Here,

1. Given that the matrix A has the Singular Value Decomposition A = UEV', where: V2 6 0 0 0 0 4 0 0 0 0 0 0 6 U = 2/2 V2 V2 (a) What is the rank r of A? (b) Write down the compact form of the SVD of A

1. Given that the matrix A has the Singular Value Decomposition A = UEV', where: V2 6 0 0 0 0 4 0 0 0 0 0 0 6 U = 2/2 V2 V2 (a) What is the rank r of A? (b) Write down the compact form of the SVD of A

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

**Singular Value Decomposition (SVD) of a Matrix**

Given that the matrix $ A $ has the Singular Value Decomposition $ A = U \Sigma V^T $, where:

\[ U = \begin{pmatrix}
\frac{2}{3} & \frac{1}{\sqrt{2}} & -\frac{\sqrt{2}}{6} \\
\frac{1}{3} & 0 & \frac{2\sqrt{2}}{3} \\
-\frac{2}{3} & \frac{1}{\sqrt{2}} & \frac{\sqrt{2}}{6}
\end{pmatrix}, \]

\[ \Sigma = \begin{pmatrix}
6 & 0 & 0 & 0 \\
0 & 4 & 0 & 0 \\
0 & 0 & 0 & 0
\end{pmatrix}, \]

\[ V = \begin{pmatrix}
\frac{1}{2} & 0 & \frac{1}{\sqrt{2}} & \frac{1}{2} \\
-\frac{1}{2} & -\frac{1}{\sqrt{2}} & 0 & \frac{1}{2} \\
\frac{1}{2} & 0 & -\frac{1}{\sqrt{2}} & \frac{1}{2} \\
-\frac{1}{2} & \frac{1}{\sqrt{2}} & 0 & \frac{1}{2}
\end{pmatrix} \]

(a) **What is the rank $ r $ of $ A $?**

(b) **Write down the compact form of the SVD of $ A $**

(c) **Find the rank 1 matrix $ A' $ that is closest to $ A $ (with respect to the Frobenius norm).**

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