Homework Help is Here – Start Your Trial Now!
Math
Advanced Math
A singular value decomposition of A is as follows: 0.5 0.5 0.5 -0.5 -0.5 0.5 0.5 -0.5 -0.5 -0.5 0.5 0.5] [5 01 0.5 0.5 -0.5 0.5 Find the least-squares solution of the linear system -1 A=UEVT Ax = b, where b â₁ Ĵ₂ = - 3 2 4 0 5 [0.6 0 0 0.8 0 0 -0.8] 0.6

A singular value decomposition of A is as follows: 0.5 0.5 0.5 -0.5 -0.5 0.5 0.5 -0.5 -0.5 -0.5 0.5 0.5] [5 01 0.5 0.5 -0.5 0.5 Find the least-squares solution of the linear system -1 A=UEVT Ax = b, where b â₁ Ĵ₂ = - 3 2 4 0 5 [0.6 0 0 0.8 0 0 -0.8] 0.6

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
See similar textbooks
icon
Related questions
Question
100%
**Least-Squares Solution Using Singular Value Decomposition (SVD)** **Singular Value Decomposition:** A singular value decomposition (SVD) of \( A \) is given as follows: \[ A = U \Sigma V^T = \begin{bmatrix} 0.5 & 0.5 & 0.5 & 0.5 \\ -0.5 & -0.5 & 0.5 & 0.5 \\ 0.5 & -0.5 & 0.5 & -0.5 \\ -0.5 & 0.5 & -0.5 & 0.5 \end{bmatrix} \begin{bmatrix} 5 & 0 \\ 0 & 5 \\ 0 & 0 \\ 0 & 0 \end{bmatrix} \begin{bmatrix} 0.6 & -0.8 \\ 0.8 & 0.6 \end{bmatrix} \] **Objective:** Find the least-squares solution of the linear system: \[ Ax = b, \text{ where } b = \begin{bmatrix} -1 \\ 3 \\ 2 \\ 4 \end{bmatrix} \] **Solutions:** \[ \hat{x}_1 = \; \_\_\_; \] \[ \hat{x}_2 = \; \_\_\_; \] By solving the system using the given SVD of the matrix \( A \), we can find the least-squares solution \( \hat{x} \). **Explanation of Graphs/Diagrams:** Here we have a matrix \( A \) decomposed into three matrices \( U \), \( \Sigma \), and \( V^T \). - \( U \) is a 4x4 orthogonal matrix. - \( \Sigma \) is a 4x2 diagonal matrix with non-negative real numbers on the diagonal (the singular values). - \( V^T \) is the transpose of a 2x2 orthogonal matrix. Using this decomposition, we find \( x \) that minimizes the equation \( Ax = b \).
Transcribed Image Text:**Least-Squares Solution Using Singular Value Decomposition (SVD)** **Singular Value Decomposition:** A singular value decomposition (SVD) of \( A \) is given as follows: \[ A = U \Sigma V^T = \begin{bmatrix} 0.5 & 0.5 & 0.5 & 0.5 \\ -0.5 & -0.5 & 0.5 & 0.5 \\ 0.5 & -0.5 & 0.5 & -0.5 \\ -0.5 & 0.5 & -0.5 & 0.5 \end{bmatrix} \begin{bmatrix} 5 & 0 \\ 0 & 5 \\ 0 & 0 \\ 0 & 0 \end{bmatrix} \begin{bmatrix} 0.6 & -0.8 \\ 0.8 & 0.6 \end{bmatrix} \] **Objective:** Find the least-squares solution of the linear system: \[ Ax = b, \text{ where } b = \begin{bmatrix} -1 \\ 3 \\ 2 \\ 4 \end{bmatrix} \] **Solutions:** \[ \hat{x}_1 = \; \_\_\_; \] \[ \hat{x}_2 = \; \_\_\_; \] By solving the system using the given SVD of the matrix \( A \), we can find the least-squares solution \( \hat{x} \). **Explanation of Graphs/Diagrams:** Here we have a matrix \( A \) decomposed into three matrices \( U \), \( \Sigma \), and \( V^T \). - \( U \) is a 4x4 orthogonal matrix. - \( \Sigma \) is a 4x2 diagonal matrix with non-negative real numbers on the diagonal (the singular values). - \( V^T \) is the transpose of a 2x2 orthogonal matrix. Using this decomposition, we find \( x \) that minimizes the equation \( Ax = b \).
Expert Solution
steps

Step by step

Solved in 3 steps with 3 images

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Similar questions
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
Basic Technical Mathematics
Basic Technical Mathematics
Advanced Math
ISBN:
9780134437705
Author:
Washington
Publisher:
PEARSON
Topology
Topology
Advanced Math
ISBN:
9780134689517
Author:
Munkres, James R.
Publisher:
Pearson,
SEE MORE TEXTBOOKS