Exercise 6.7.21

Here we use the notation in 6.26.

1.

Compute

$$T{T}^{\text{†}}T(v_i)=T{T}^{\text{†}}({\sigma }_i{u}_i)=T(v_i)$$

if $i\le r$, while

$$T{T}^{\text{†}}T(v_i)=T{T}^{\text{†}}(0)=0=T(v_i)$$

if $i>r$.

2.

Compute

$$T^{\text{†}}T{T}^{\text{†}}(u_i)=T^{\text{†}}T(\frac{1}{{\sigma }_i}{v}_i)=T^{\text{†}}(u_i)$$

if $i\le r$, while

$$T^{\text{†}}T{T}^{\text{†}}(u_i)=T^{\text{†}}T(0)=0=T^{\text{†}}(u_i)$$

if $i>r$.

3.

Pick an orthonormal basis $\alpha$. Let $A$ be the matrix ${[T]}_{\alpha }$ and $A=U\mathrm{\Sigma }{V}^{\text{∗}}$ be a singular value decomposition of $A$. Then we have

$${(A^{\text{†}}A)}^{\text{∗}}={(V{\mathrm{\Sigma }}^{\text{†}}{U}^{\text{∗}}U\mathrm{\Sigma }{V}^{\text{∗}})}^{\text{∗}}={(V{\mathrm{\Sigma }}^{\text{†}}\mathrm{\Sigma }{V}^{\text{∗}})}^{\text{∗}}=A^{\text{†}}A$$

and

$${(A{A}^{\text{†}})}^{\text{∗}}={(U\mathrm{\Sigma }{V}^{\text{∗}}V{\mathrm{\Sigma }}^{\text{†}}{U}^{\text{∗}})}^{\text{∗}}={(U\mathrm{\Sigma }{\mathrm{\Sigma }}^{\text{†}}{U}^{\text{∗}})}^{\text{∗}}=A{A}^{\text{†}}.$$