Review Questions for Singular Value Decompositions

  1. For a matrix \({\bf A}\) with SVD decomposition \({\bf A} = {\bf U \Sigma V}^T\), what are the columns of \({\bf U}\) and how can we find them? What are the columns of \({\bf V}\) and how can we find them? What are the entries of \({\bf \Sigma}\) and how can we find them?
  2. What special properties are true of \({\bf U}\), \({\bf V}\) and \({\bf \Sigma}\)?
  3. What are the shapes of \({\bf U}\), \({\bf V}\) and \({\bf \Sigma}\) in the full SVD of an \(m \times n\) matrix?
  4. What are the shapes of \({\bf U}\), \({\bf V}\) and \({\bf \Sigma}\) in the reduced SVD of an \(m \times n\) matrix?
  5. What is the cost of computing the SVD?
  6. Given an already computed SVD of a matrix \({\bf A}\), what is the cost of using the SVD to solve a linear system \({\bf A}\bf{x} = \bf{b}\)? How would you use the SVD to solve this system?
  7. How do you use the SVD to compute a low-rank approximation of a matrix? For a small matrix, you should be able to compute a given low rank approximation (i.e. rank-one, rank-two).
  8. Given the SVD of a matrix \({\bf A}\), what is the SVD of \({\bf A}^+\) (the psuedoinverse of \({\bf A}\))?
  9. Given the SVD of a matrix \({\bf A}\), what is the 2-norm of the matrix? What is the 2-norm condition number of the matrix?