Exercise 6.7.1

Answers

1.: No. The mapping from $ℝ^{2}$ to $ℝ$ has no eigenvalues.
2.: No. It’s the the positive square root of the eigenvalues of $A^{∗} A$ . For example, the singular value of $2 I_{2}$ is $2, 2$ but not eigenvalues of ${(2 I)}^{∗} (2 I)$ , which is $4, 4$ .
3.: Yes. The eigenvalue of $A^{∗} A$ is $σ^{2}$ . And the singular value of $cA$ is the positive square root of the eigenvalue of ${(cA)}^{∗} (cA) = | c |^{2} A^{∗} A$ , which is $| c |^{2} σ^{2}$ . So the singular value of $cA$ is $| c | σ$ .
4.: Yes. This is the definition.
5.: No. For example, the singular value of $2 I_{2}$ is $2, 2$ but not eigenvalues of ${(2 I)}^{∗} (2 I)$ , which is $4, 4$ .
6.: No. If $Ax = b$ is inconsistent, then $A^{†} b$ could never be the solution.
7.: Yes. The definition is well-defined.

jephianlin

2011-06-27 00:00