Exercise 2.2.5

Answers

1.
If $A$ has a rank $r = n$ , then there are $r$ nonzeros in $Σ$ , these are the singular values.
1.
When $A$ has independent columns, we know that $A^{A}$ is invertible. So we have $Σ = U^{T} AV$ and $Σ^{T} Σ = V^{T} A^{T} U U^{T} AV = V^{T} A^{T} AV$ , the inverse of $Σ^{T} Σ$ is ${(Σ^{T} Σ)}^{- 1} = V^{T} {(A^{T} A)}^{- 1} V$ since $(Σ^{T} Σ) {(Σ^{T} Σ)}^{- 1} = V^{T} A^{T} AV V^{T} {(A^{T} A)}^{- 1} V = I$
1.
We let $Σ^{+} = {(Σ^{T} Σ)}^{- 1} Σ^{T} = V^{T} {(A^{T} A)}^{- 1} V V^{T} A^{T} U = V^{T} {(A^{T} A)}^{- 1} A^{T} U$
1.
Then we have $A^{+} = {(A^{T} A)}^{- 1} A^{T} = {(V Σ^{2} V^{T})}^{- 1} V Σ U^{T} = V Σ^{- 2} V^{T} V Σ U^{T} = V Σ^{- 1} U^{T}$

niuers

2020-03-20 00:00