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