Obtain full V matrix from Eigen SVD function I'm using the eigen library to calculate the fundamental matrix from two cameras using the 8 point algorithm. The 8 point algorithm needs the last column of the V matrix.
Comparing the results to matlab I discovered that Eigen's SVD function does not return an 8x9 V matrix (n-by-p) as matlab does but instead an 8x8 one where the 2 last columns have different values. 
The Eigen documentation says "You can ask for only thin U or V to be computed, meaning the following. In case of a rectangular n-by-p matrix, letting m be the smaller value among n and p, there are only m singular vectors; the remaining columns of U and V do not correspond to actual singular vectors. Asking for thin U or V means asking for only their m first columns to be formed. So U is then a n-by-m matrix, and V is then a p-by-m matrix."
Does anyone know how to get the missing column from Eigen?
 A: The issue here is the orientation of the axes for the nullspace $\mathcal{N}\left( \mathbf{A} \right)$.
Consider the matrix 
$$
\mathbf{A} = 
\left[
\begin{array}{ccc}
  0 & 0 & 1\\
  0 & 0 & 2
\end{array}
\right].
$$
The nullspace $\mathcal{N}\left( \mathbf{A} \right)$ is the $x-y$ plane. We have many ways to orient that space 
$$
  \mathcal{N}\left( \mathbf{A} \right) =
\text{span } \left\{ \,
  \left[
    \begin{array}{c}
      1 \\ 0 \\ 0
    \end{array}
  \right], \,
  \left[
    \begin{array}{c}
      0 \\ 1 \\ 0
    \end{array}
  \right]\, \right\} =
\text{span } \left\{ \frac{1}{\sqrt{2}}
  \left[
    \begin{array}{c}
      1 \\ 1 \\ 0
    \end{array}
  \right], \frac{1}{\sqrt{2}}
 \left[    \begin{array}{r}
      1 \\ -1 \\ 0
    \end{array}
  \right] \,
\right\}.
$$
In fact, the general cases include reflections and rotations like
$$
  \mathcal{N}\left( \mathbf{A} \right) =
\text{span } \left\{ \,
  \left[
    \begin{array}{c}
      \cos \theta \\ \sin \theta \\ 0
    \end{array}
  \right], \,
  \left[
    \begin{array}{r}
      -\sin \theta \\ \cos \theta \\ 0\phantom{-}
    \end{array}
  \right]\, \right\}, \quad
0 \le \theta \lt 2\pi.
$$
