Review Questions for Singular Value Decompositions

1. For a matrix $\mathbf{A}$ with SVD decomposition $\mathbf{A}={\mathbf{U}\mathbf{\Sigma }\mathbf{V}}^T$, what are the columns of $\mathbf{U}$ and how can we find them? What are the columns of $\mathbf{V}$ and how can we find them? What are the entries of $\mathbf{\Sigma }$ and how can we find them?
2. What special properties are true of $\mathbf{U}$, $\mathbf{V}$ and $\mathbf{\Sigma }$?
3. What are the shapes of $\mathbf{U}$, $\mathbf{V}$ and $\mathbf{\Sigma }$ in the full SVD of an $m\times n$ matrix?
4. What are the shapes of $\mathbf{U}$, $\mathbf{V}$ and $\mathbf{\Sigma }$ 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 $\mathbf{A}$, what is the cost of using the SVD to solve a linear system $\mathbf{A}\mathbf{x}\mathbf{=}\mathbf{b}$? 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 $\mathbf{A}$, what is the SVD of ${\mathbf{A}}^{+}$ (the psuedoinverse of $\mathbf{A}$)?
9. Given the SVD of a matrix $\mathbf{A}$, what is the 2-norm of the matrix? What is the 2-norm condition number of the matrix?