A9_soln - Math 235 Assignment 9 Solutions 1 Let A = 1 2 3 1...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Math 235 Assignment 9 Solutions 1. Let A = 1- 2 3 1 2 . Find the maximum and minimum value of k A~x k for ~x ∈ R 2 subject to the constraint k ~x k = 1. Solution: We have A T A = 11 0 8 . Hence, the eigenvalues of A T A are λ 1 = 11 and λ 2 = 8. Thus the maximum of k A~x k subject to k ~x k = 1 is √ 11 and the minimum is √ 8. 2. Find a singular value decomposition of each of the following matrices. a) 1 1- 1 1 Solution: Let A = 1 1- 1 1 then A T A = 2 0 0 2 . Thus the only eigenvalue of A T A is λ 1 = 2 with multiplicity 2, and A is orthogonally diagonalized by V = 1 0 0 1 . The singular values of A are σ 1 = √ 2 and σ 2 = √ 2. Thus the matrix Σ is Σ = √ 2 √ 2 . Next compute ~u 1 = 1 σ 1 A~v 1 = 1 √ 2 1 1- 1 1 1 = 1- 1 ~u 2 = 1 σ 2 A~v 2 = 1 √ 2 1 1- 1 1 1 = 1 1 Thus we have U = 1 / √ 2 1 / √ 2- 1 / √ 2 1 / √ 2 . Then A = U Σ V T as required. 2 b) 1- 4- 2 2 2 4 Solution: Let A = 1- 4- 2 2 2 4 then A T A = 9 0 36 and the eigenvalues of...
View Full Document

{[ snackBarMessage ]}

Page1 / 4

A9_soln - Math 235 Assignment 9 Solutions 1 Let A = 1 2 3 1...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online