Exercise 6.10.13

Answers

1.

λ_{1}

is the largest eigenvalue of

A^{∗} A

, then we know that

σ_{i} = \sqrt{λ_{i}} = ∥ A ∥

2.

This comes from that the nonzero singular values of

A^{†}

are

σ_{r}^{- 1} \geq σ_{r - 1}^{- 1} \geq \dots \geq σ_{1}^{- 1} .

3.

A

is invertible with the largest and the smallest eigenvalues of

A^{∗} A

to be

λ_{1}

and

λ_{n} > 0

, we know that

σ_{1} = \sqrt{λ_{1}}

and

σ_{n} = \sqrt{λ_{n}}

. Hence we have

cond (A) = \frac{σ_{1}}{σ_{n}} .

jephianlin

2011-06-27 00:00