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 ${\bf A}^{-1}$?