<s>[INST] Unable to calculate pseudo-inverse $A^TA$ I'm trying to calculate the pseudo-inverse of $A^TA$ as described in this paper:

The SVD is particularly simple to calculate when
  the matrix is of the form $A^TA$ because $U=V$ and the rows of $U$
  are the eigenvectors of $A^TA$ and the singular values in $D$ are the
  eigenvalues of $A^TA$.
  Since $U$ and $V$ are orthogonal matrices, the inverse of $M$ is then
  $M^{-1}=V^TD^{-1}U$

Since the pseudo-inverse of a non-singular matrix is it's inverse I tried to do this with a non-singular matrix and see if I get the inverse. I tried it with the following:
$A^TA = \begin{pmatrix}
90 & 51 \\
51 & 29
\end{pmatrix}$ where $A = \begin{pmatrix}
3 & 2 \\
9 & 5
\end{pmatrix}$
Eigenvalues and eigenvectors:
$\lambda_1 = 118.9243;$  $\lambda_2=0.0757;$ $v_1 = \begin{pmatrix}
1 \\
0.5671
\end{pmatrix};$ $v_2 = \begin{pmatrix}
1 \\
-1.7632
\end{pmatrix}$
$U = V=\begin{pmatrix}
1 & 0.5671 \\
1 & -1.7632
\end{pmatrix}$
$D = \begin{pmatrix}
118.9243 & 0\\
0 & 0.0757
\end{pmatrix}$
And my result is:
$V^TD^{-1}U= \begin{pmatrix}
13.22 & -23.29 \\
-23.29 & 41.07
\end{pmatrix} \neq (A^TA)^{-1}$ 
Can someone help me? What am I doing wrong?
 [/INST] Pick up the thread at the eigenvectors or the product matrix. The ordering was mixed. It should be:
$$
  v_{1} = 
\left[
\begin{array}{c}
 1.76322 \\
 1 \\
\end{array}
\right], 
\qquad
  v_{2} = 
\left[
\begin{array}{c}
 -0.567144 \\
 1 \\
\end{array}
\right].
$$
Normalized, these are the column vectors of the domain matrix:
$$
\mathbf{V} =
\left[
\begin{array}{cc}
 0.869844 & -0.493327 \\
 0.493327 & 0.869844 \\
\end{array}
\right]
$$
Next, $\mathbf{U} \ne \mathbf{V}!$:
$$
\mathbf{A} =   \mathbf{U} \, \Sigma \, \mathbf{V}^{T} 
\qquad \Rightarrow \qquad
\mathbf{U} = \mathbf{A} \, \mathbf{V} \, \Sigma^{-1}.
$$
Now perform your cross check:
$$
\begin{align}
%
\mathbf{A} &= \mathbf{U} \, \Sigma \, \mathbf{V}^{T} \\
%
\left[
\begin{array}{cc}
 3 & 2 \\
 9 & 5 \\
\end{array}
\right]
%
&=
% U
\left[
\begin{array}{cr}
 0.329767 & 0.944062 \\
 0.944062 & -0.329767 \\
\end{array}
\right]
% Sigma
\left[
\begin{array}{cc}
 10.9052 & 0. \\
 0. & 0.275097 \\
\end{array}
\right]
% V^T
\left[
\begin{array}{rc}
 0.869844 & 0.493327 \\
 -0.493327 & 0.869844 \\
\end{array}
\right]
%
\end{align}
$$
All null spaces are trivial and the classic inverse is the pseudoinverse:
$$
  \mathbf{A}^{-1} = \mathbf{A}^{\dagger} = \mathbf{V} \, \Sigma^{-1} \, \mathbf{U}^{T} =
\frac{1}{3}
\left[
\begin{array}{rr}
 -5 & 2 \\
 9 & -3 \\
\end{array}
\right].
$$
</s>