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Problem: Singular Value Decomposition and Advanced Matrix Concepts Let A ER4×4 be the matrix given by: /1 200 2 0 3 0 A = 0 3 0 1 0 0 1 1 (a) Compute the Singular Value Decomposition (SVD) of matrix A. Provide the following components of the SVD: The left singular matrix U, • The diagonal matrix of singular values Σ, The right singular matrix V. Explain the process of obtaining each component of the SVD, including intermediate steps. (b) Determine the eigenvalues and eigenvectors of ATA and AAT. • Show the relationship between the eigenvalues of ATA and the singular values of A. Verify that the eigenvectors of AT A correspond to the right singular vectors of A, and that the eigenvectors of AAT correspond to the left singular vectors of A. (c) Rank and approximation of A. • • • Calculate the rank of A using the SVD. Explain how the rank of a matrix can be derived from its singular values. Using the SVD, find the best rank-2 approximation A2 of matrix A. Discuss how the approximation is obtained by truncating the smaller singular values. Compute the Frobenius norm of the difference ||A - A2||F and interpret the result. (d) Low-rank matrix compression and energy capture. • Suppose you are using the SVD for data compression. How much energy (i.e., variance) is captured by the best rank-2 approximation of A? • Compute the energy captured by the rank-2 approximation as a percentage of the total energy, where the energy is defined as the sum of squares of the singular values.

Problem: Singular Value Decomposition and Advanced Matrix Concepts Let A ER4×4 be the matrix given by: /1 200 2 0 3 0 A = 0 3 0 1 0 0 1 1 (a) Compute the Singular Value Decomposition (SVD) of matrix A. Provide the following components of the SVD: The left singular matrix U, • The diagonal matrix of singular values Σ, The right singular matrix V. Explain the process of obtaining each component of the SVD, including intermediate steps. (b) Determine the eigenvalues and eigenvectors of ATA and AAT. • Show the relationship between the eigenvalues of ATA and the singular values of A. Verify that the eigenvectors of AT A correspond to the right singular vectors of A, and that the eigenvectors of AAT correspond to the left singular vectors of A. (c) Rank and approximation of A. • • • Calculate the rank of A using the SVD. Explain how the rank of a matrix can be derived from its singular values. Using the SVD, find the best rank-2 approximation A2 of matrix A. Discuss how the approximation is obtained by truncating the smaller singular values. Compute the Frobenius norm of the difference ||A - A2||F and interpret the result. (d) Low-rank matrix compression and energy capture. • Suppose you are using the SVD for data compression. How much energy (i.e., variance) is captured by the best rank-2 approximation of A? • Compute the energy captured by the rank-2 approximation as a percentage of the total energy, where the energy is defined as the sum of squares of the singular values.

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter4: Eigenvalues And Eigenvectors
Section4.5: Iterative Methods For Computing Eigenvalues
Problem 51EQ
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For the SVD, eigenvalue decompositions, and matrix approximations, you may use appropriate computational tools for complex calculations, but all theoretical steps and justifications must be clearly presented.

Problem: Singular Value Decomposition and Advanced Matrix Concepts Let A ER4×4 be the matrix given by: /1 200 2 0 3 0 A = 0 3 0 1 0 0 1 1 (a) Compute the Singular Value Decomposition (SVD) of matrix A. Provide the following components of the SVD: The left singular matrix U, • The diagonal matrix of singular values Σ, The right singular matrix V. Explain the process of obtaining each component of the SVD, including intermediate steps. (b) Determine the eigenvalues and eigenvectors of ATA and AAT. • Show the relationship between the eigenvalues of ATA and the singular values of A. Verify that the eigenvectors of AT A correspond to the right singular vectors of A, and that the eigenvectors of AAT correspond to the left singular vectors of A.
Transcribed Image Text:Problem: Singular Value Decomposition and Advanced Matrix Concepts Let A ER4×4 be the matrix given by: /1 200 2 0 3 0 A = 0 3 0 1 0 0 1 1 (a) Compute the Singular Value Decomposition (SVD) of matrix A. Provide the following components of the SVD: The left singular matrix U, • The diagonal matrix of singular values Σ, The right singular matrix V. Explain the process of obtaining each component of the SVD, including intermediate steps. (b) Determine the eigenvalues and eigenvectors of ATA and AAT. • Show the relationship between the eigenvalues of ATA and the singular values of A. Verify that the eigenvectors of AT A correspond to the right singular vectors of A, and that the eigenvectors of AAT correspond to the left singular vectors of A.
(c) Rank and approximation of A. • • • Calculate the rank of A using the SVD. Explain how the rank of a matrix can be derived from its singular values. Using the SVD, find the best rank-2 approximation A2 of matrix A. Discuss how the approximation is obtained by truncating the smaller singular values. Compute the Frobenius norm of the difference ||A - A2||F and interpret the result. (d) Low-rank matrix compression and energy capture. • Suppose you are using the SVD for data compression. How much energy (i.e., variance) is captured by the best rank-2 approximation of A? • Compute the energy captured by the rank-2 approximation as a percentage of the total energy, where the energy is defined as the sum of squares of the singular values.
Transcribed Image Text:(c) Rank and approximation of A. • • • Calculate the rank of A using the SVD. Explain how the rank of a matrix can be derived from its singular values. Using the SVD, find the best rank-2 approximation A2 of matrix A. Discuss how the approximation is obtained by truncating the smaller singular values. Compute the Frobenius norm of the difference ||A - A2||F and interpret the result. (d) Low-rank matrix compression and energy capture. • Suppose you are using the SVD for data compression. How much energy (i.e., variance) is captured by the best rank-2 approximation of A? • Compute the energy captured by the rank-2 approximation as a percentage of the total energy, where the energy is defined as the sum of squares of the singular values.
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