SVD decomposition representation Let $x=A^T(AA^T)^{-1}b$ given that $A_{m\times n}$ is a full row rank matrix and $b$ is a vector. By $SVD$ decomposition $A=U\Sigma V^T$. 
How can I express $x$ with $U,V,b$ and the addition of the partial matrix $\Sigma_{m\times m}$ which contains the non zeroes segments of $\Sigma$.
Also Can It be written with a slimmer version using $u_i\in \mathbb R^{m\times1},v_i\in \mathbb R^{n\times1}$, vectors from $U,V$ and $\sigma_i$ the diagonal values of $\Sigma$?
Is the following valid\good enough? 
$$x=A^T(AA^T)^{-1}b=(UΣV^T)^T(UΣV(UΣV^T)^T)^{-1}b=U^TΣ^TV(UΣVU^TΣ^TV)^{-1}b$$
$$=U^T ΣV(UΣVU^T ΣV)^{-1}b$$
 A: I assume that $m \le n$ holds? Then note:
$$AA^t = U \Sigma V^t V \Sigma^t U^t = U \Sigma \Sigma^t U^t$$
Since $A$ has full rank, $\Sigma \Sigma^t$ also has full rank. From $U \Sigma \Sigma^t U^t U (\Sigma\Sigma^t)^{-1}U = I$, we can infer $(AA^t)^{-1}$ = $U(\Sigma\Sigma^t)^{-1}U^t$. This yields:
$$A^t(AA^t)^{-1} = V \Sigma^t U^t U (\Sigma\Sigma^t)^{-1} U^t = V \Sigma^t (\Sigma \Sigma^t)^{-1} U^t.$$
We have $\Sigma = (\Sigma_{m \times m})$ and therefore $\Sigma \Sigma^t = \Sigma_{m \times m} \Sigma_{m \times m}^t = \Sigma_{m \times m}^2$. This implies $$\Sigma^t(\Sigma \Sigma^t)^{-1} = \begin{pmatrix}\Sigma_{m \times m}^{-1} \\ 0\end{pmatrix}.$$
Edit: If $u_i$ resp. $v_i$ are the rows of $U$ resp. $V$, you can simplify this further.
$$\begin{pmatrix}v_1 \\ v_2 \\ \vdots \\v_n\end{pmatrix} \begin{pmatrix}\Sigma_{m \times m}^{-1} \\ 0 \end{pmatrix} \begin{pmatrix}u_1^t & u_2^t & \cdots & u_m^t\end{pmatrix} =\begin{pmatrix}v_1 \\ v_2 \\ \vdots \\v_n\end{pmatrix} \begin{pmatrix}\Sigma_{m \times m}^{-1} u_1^t & \Sigma_{m \times m}^{-1} u_2^t & \cdots & \Sigma_{m \times m}^{-1}u_m^t \\ 0 & 0 & \cdots & 0\end{pmatrix}$$
Now if $\sigma_i$ are the diagonal elements of $\Sigma_{m \times m}$, then the $(i, j)$-th coordinate of this matrix is given by $\sum \limits_{k = 1}^m v_{i, k} \sigma_{k, k}^{-1} u_{j, k}$.
A: The target matrix has full row rank
$$
\mathbf{A} \in \mathbb{C}^{m\times n}_{m},
$$
and $m < n$. The null space $\mathcal{N}\left( \mathbf{A}^{*} \right)$ is trivial and the singular value decomposition takes the form
$$
\begin{align}
  \mathbf{A} &=
  \mathbf{U} \, \Sigma \, \mathbf{V}^{*} \\
%
 &=
% U 
  \left[ \begin{array}{c}
     \color{blue}{\mathbf{U}_{\mathcal{R}}}
  \end{array} \right]  
% Sigma
  \left[ \begin{array}{cccccc}
     \sigma_{1} & 0 & \dots &  &   & \dots &  0 \\
     0 & \sigma_{2}  &&&&&  \\
     \vdots && \ddots &&&& \vdots\\
      0 & & & \sigma_{m} & 0 & \dots & 0 \\
  \end{array} \right]
% V 
  \left[ \begin{array}{c}
     \color{blue}{\mathbf{V}^{*}_{\mathcal{R}}} \\
     \color{red}{\mathbf{V}^{*}_{\mathcal{N}}}
  \end{array} \right]  \\[3pt]
%
  & =
% U 
  \left[ \begin{array}{cc}
     \color{blue}{\mathbf{U}_{\mathcal{R}}} 
  \end{array} \right]  
% Sigma
  \left[ \begin{array}{cc}
     \mathbf{S}_{m\times m} & \mathbf{0} \\
  \end{array} \right]
% V 
  \left[ \begin{array}{c}
     \color{blue}{\mathbf{V}^{*}_{\mathcal{R}}} \\
     \color{red}{\mathbf{V}^{*}_{\mathcal{N}}}
  \end{array} \right]  \\[3pt]
%
\end{align}
$$
The product matrix is
$$
  \mathbf{A} \, \mathbf{A}^{*} =
  \left( \mathbf{U} \, \Sigma \, \mathbf{V}^{*} \right)
  \left( \mathbf{V} \, \Sigma \, \mathbf{U}^{*} \right) =
  \color{blue}{\mathbf{U}_{\mathcal{R}}} \, \mathbf{S}^{2} \, \color{blue}{\mathbf{U}^{*}_{\mathcal{R}}} 
$$
with the inverse being
$$
  \left( \mathbf{A} \, \mathbf{A}^{*} \right)^{-1} =
  \color{blue}{\mathbf{U}_{\mathcal{R}}} \, \mathbf{S}^{-2} \, \color{blue}{\mathbf{U}^{*}_{\mathcal{R}}} 
$$
Finally
$$
\begin{align}
\mathbf{A}^{*} \left( \mathbf{A} \, \mathbf{A}^{*} \right)^{-1} &= 
\left( 
% V 
  \left[ \begin{array}{cc}
     \color{blue}{\mathbf{V}_{\mathcal{R}}} &
     \color{red}{\mathbf{V}_{\mathcal{N}}}
  \end{array} \right]  
% Sigma
  \left[ \begin{array}{c}
     \mathbf{S} \\ \mathbf{0} \\
  \end{array} \right]
% U 
     \color{blue}{\mathbf{U}^{*}_{\mathcal{R}}} 
\right)
\color{blue}{\mathbf{U}_{\mathcal{R}}} \, \mathbf{S}^{-2} \, \color{blue}{\mathbf{U}^{*}_{\mathcal{R}}} \\
&=
% V 
  \left[ \begin{array}{cc}
     \color{blue}{\mathbf{V}_{\mathcal{R}}} &
     \color{red}{\mathbf{V}_{\mathcal{N}}}
  \end{array} \right]  
% Sigma
  \left[ \begin{array}{c}
     \mathbf{S}^{-1} \\ \mathbf{0} \\[2pt]
  \end{array} \right]
% U 
     \color{blue}{\mathbf{U}_{\mathcal{R}}}  \\
&=
\mathbf{A}^{\dagger}
\end{align}
$$
