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Unformatted text preview: EE263 S. Lall 2011.05.17.01 Midterm Exam Spring 2010-2011 Solutions This is a 24-hour take-home exam. You may use any books, lecture-notes, or computer programs that you wish, but you may not discuss this exam with anyone until Tuesday May 10, after everyone has taken it. The only exception is that you can ask the TA’s or Sanjay Lall for clarification. Web page for typos. The web page http://junction.stanford.edu/~lall/ee263/midterm should always contain all known typos; check this before contacting one of us for clarification. Please note that we have tried pretty hard to make the exam unambiguous and clear, so we are unlikely to say much. If you choose to send out an email for clarification, please use the staff email address [email protected] so that you can get reply as soon as possible. Since you have 24 hours, we expect your solutions to be legible, neat, and clear. Do not hand in your rough notes, and please try to simplify your solutions as much as possible. Good Luck! 1. Square matrices and the SVD. 60003 Let A be an n × n real matrix. State whether each of the following statements is true or false. Do not give any explanation or show any work. (a) If x is an eigenvector of A , then x is either a left or right singular vector of A (b) If λ is an eigenvalue of A , then | λ | is a singular value (c) If A is symmetric, then every singular value of A is also an eigenvalue of A (d) If A is symmetric, then every singular vector of A is also an eigenvector of A (e) If A is symmetric with the following singular value decomposition A = U Σ V T then U = V (f) If A is invertible, then σ i negationslash = 0 for all i = 1 ,...,n Solution. (a) False .Consider the matrix A = bracketleftbigg 1 1 1 bracketrightbigg The SVD of A is U = bracketleftbigg . 8507 − . 5257 . 5257 . 8507 bracketrightbigg Σ = bracketleftbigg 1 . 618 . 618 bracketrightbigg V = bracketleftbigg . 5257 − . 8507 . 8507 . 5257 bracketrightbigg bracketleftbigg 1 bracketrightbigg is an eigenvector of A but is neither a right nor a left singular vector of A. (b) False . For the matrix mentioned in (a), the eigenvalues are 1,1 while the singular values are 1.618,0.618 (c) False . Singular values are non-negative for any matrix but the eigenvalues, even if the matrix is symmetric, can be negative. 1 EE263 S. Lall 2011.05.17.01 (d) True . If A is symmetric, it has the following eigenvalue decomposition A = Q Λ Q T where Q is orthogonal. For all the negative eigenvalues in Λ, if we retain their magnitude in Λ and multiply the corresponding eigenvectors in Q (or Q T ) by -1, we get the SVD of A (since Q was orthogonal). Therefore every singular vector of A is also an eigenvector of A....
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This note was uploaded on 08/06/2011 for the course EE 263 taught by Professor Boyd,s during the Summer '08 term at Stanford.
- Summer '08