Homepage › Solution manuals › Stephen Friedberg › Linear Algebra › Exercise 6.7.20

Exercise 6.7.20

Answers

Let $A = U Σ V^{∗}$ be a singular value decomposition of $A$ . Then we have

O = A^{2} = U Σ V^{∗} U Σ V,

which means

Σ V^{∗} U Σ = O

since $U$ and $V$ are invertible. Now let ${σ_{i}}_{i = 1}^{r}$ is the set of those singular values of $A$ . Denote $D$ to be the diagonal matrix with $D_{ii} = \frac{1}{σ^{2}}$ if $i \leq r$ while $D_{ii} = 1$ if $i > r$ . Then we have $Σ D = D Σ = Σ^{†}$ . This means that

Σ^{†} V^{∗} U Σ^{†} = D Σ V^{∗} U Σ D = DOD = O .

Now we have

{(A^{†})}^{2} = {(V Σ^{†} U^{∗})}^{2} = V Σ^{†} U^{∗} V Σ^{†} U^{∗}

= V {(Σ^{†} V^{∗} U Σ^{†})}^{∗} U^{∗} = V O U^{∗} = O .

jephianlin

2011-06-27 00:00

Exercise 6.7.20

Answers

Comments

Add answer