Consider a set of algebraic equations Mx =b where -0.778 -1.222 -0.889 2.111 1 M=0 0.889 -0.111 The singular value decomposition of M is M = USV* where 0.613 0.577] -0.540 U=-0.26O -0.774 0.577 0.800 0.161 0.577 [2.91 1.30 0 -0.461 0.348 -0.816 -0.912 -0.408 -0.408 V = 0.034 0.887 0.216 1) Based on this decomposition, show that if an exact solution to Mx = b exists, then it will not be unique. (ii) Determine any value for the vector b such that a solution to Mx = b will exist?
Consider a set of algebraic equations Mx =b where -0.778 -1.222 -0.889 2.111 1 M=0 0.889 -0.111 The singular value decomposition of M is M = USV* where 0.613 0.577] -0.540 U=-0.26O -0.774 0.577 0.800 0.161 0.577 [2.91 1.30 0 -0.461 0.348 -0.816 -0.912 -0.408 -0.408 V = 0.034 0.887 0.216 1) Based on this decomposition, show that if an exact solution to Mx = b exists, then it will not be unique. (ii) Determine any value for the vector b such that a solution to Mx = b will exist?
Advanced Engineering Mathematics
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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 and Solutions to Linear Systems**
**Problem 3.17:**
Consider a set of algebraic equations \(Mx = b\) where
\[
M = \begin{bmatrix}
1 & -0.778 & -1.222 \\
0 & 0.889 & -0.889 \\
1 & -0.111 & 2.111
\end{bmatrix}
\]
The singular value decomposition of \(M\) is \(M = USV^*\) where
\[
U = \begin{bmatrix}
-0.540 & 0.613 & 0.577 \\
-0.260 & -0.774 & 0.577 \\
0.800 & 0.161 & 0.577
\end{bmatrix}
\]
\[
S = \begin{bmatrix}
2.91 & 0 & 0 \\
0 & 1.30 & 0 \\
0 & 0 & 0
\end{bmatrix}
\]
\[
V = \begin{bmatrix}
-0.461 & 0.348 & -0.816 \\
0.034 & -0.912 & -0.408 \\
0.887 & 0.216 & -0.408
\end{bmatrix}
\]
(i) Based on this decomposition, show that if an exact solution to \(Mx = b\) exists, then it will not be unique.
(ii) Determine any value for the vector \(b\) such that a solution to \(Mx = b\) will exist.
(iii) What is the set of all \(b\) such that a solution to \(Mx = b\) will exist?
(iv) For the \(b\) of part (ii), determine a solution to \(Mx = b\).
(v) Using the \(b\) of part (ii), determine all solutions to \(Mx = b\). Rather than performing the calculations based on the given numbers, use the notation below to develop a formula for the solution.
\[
U = [u_1, u_2, u_3] \quad S = \begin{bmatrix} \sigma_1 & 0 & 0 \\ 0 & \sigma_2 & 0 \\ 0 & 0 & 0 \end{bmatrix}](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F65a54057-de91-4407-81dc-84071bb6029d%2F0091b460-edb7-45a7-8b9f-258eb41507bf%2F3pn696f_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Singular Value Decomposition and Solutions to Linear Systems**
**Problem 3.17:**
Consider a set of algebraic equations \(Mx = b\) where
\[
M = \begin{bmatrix}
1 & -0.778 & -1.222 \\
0 & 0.889 & -0.889 \\
1 & -0.111 & 2.111
\end{bmatrix}
\]
The singular value decomposition of \(M\) is \(M = USV^*\) where
\[
U = \begin{bmatrix}
-0.540 & 0.613 & 0.577 \\
-0.260 & -0.774 & 0.577 \\
0.800 & 0.161 & 0.577
\end{bmatrix}
\]
\[
S = \begin{bmatrix}
2.91 & 0 & 0 \\
0 & 1.30 & 0 \\
0 & 0 & 0
\end{bmatrix}
\]
\[
V = \begin{bmatrix}
-0.461 & 0.348 & -0.816 \\
0.034 & -0.912 & -0.408 \\
0.887 & 0.216 & -0.408
\end{bmatrix}
\]
(i) Based on this decomposition, show that if an exact solution to \(Mx = b\) exists, then it will not be unique.
(ii) Determine any value for the vector \(b\) such that a solution to \(Mx = b\) will exist.
(iii) What is the set of all \(b\) such that a solution to \(Mx = b\) will exist?
(iv) For the \(b\) of part (ii), determine a solution to \(Mx = b\).
(v) Using the \(b\) of part (ii), determine all solutions to \(Mx = b\). Rather than performing the calculations based on the given numbers, use the notation below to develop a formula for the solution.
\[
U = [u_1, u_2, u_3] \quad S = \begin{bmatrix} \sigma_1 & 0 & 0 \\ 0 & \sigma_2 & 0 \\ 0 & 0 & 0 \end{bmatrix}
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