# 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?