Review Questions for Condition Numbers

1. What is the definition of a condition number?
2. What is the condition number of solving \({\bf A}\mathbf{x} = \mathbf{b}\)?
3. What is the condition number of matrix-vector multiplication?
4. Calculate the \(p\)-norm condition number of a matrix for a given \(p\).
5. Do you want a small condition number or a large condition number?
6. What is the condition number of an orthogonal matrix?
7. If you have \(p\) accurate digits in \({\bf A}\) and \(\mathbf{b}\), how many accurate digits do you have in the solution of \({\bf A}\mathbf{x} = \mathbf{b}\) if the condition number of \({\bf A}\) is \(\kappa\)?
8. When solving a linear system \({\bf A}\mathbf{x} = \mathbf{b}\), does a small residual guarantee an accurate result?
9. Consider solving a linear system \({\bf A}\mathbf{x} = \mathbf{b}\). When does Gaussian elimination with partial pivoting produce a small residual?
10. How does the condition number of a matrix \({\bf A}\) relate to the condition number of \({\bf A}^{-1}\)?