\documentclass {article}

\newcommand\reals{\mathbf{R}}
\begin{document}
\paragraph {Projection on the kernel of a matrix $J$}
$$
J\in \reals^{m\times n}
$$
SVD decomposition
\begin{eqnarray*}
J &=& U S V^T \ \ \ U\in O(m)\ \ \ V\in O(n)\\
J^{+} &=& V S^{+} U^T \\
J^{+} J &=& V S^{+} S V^T \\
&=& V \left(\begin{array} {cc}
I_m & O_{m\times n-m} \\
O_{n-m\times m} & O_{n-m\times n-m}
\end{array}\right) V^T
\end{eqnarray*}
where $m$ is the (full) rank of $J$.
\begin{eqnarray*}
I_{n} - J^{+} J &=& V \left(\begin{array} {cc}
O_m & O_{m\times n-m} \\
O_{n-m\times m} & I_{n-m}
\end{array}\right) V^T \\
&=& \left(\begin{array}{cc}
V_{1} & V_{0} \end{array}\right)
\left(\begin{array} {cc}
O_m & O_{m\times n-m} \\
O_{n-m\times m} & I_{n-m}
\end{array}\right)
\left(\begin{array}{c}
V^T_{1} \\
V^T_{0} \end{array}\right)\\
&=&
V_0 V_0^T
\end{eqnarray*}

\paragraph {Projection on the kernel of a matrix $J$}
$$
J\in \reals^{m\times n}
$$
\begin{eqnarray*}
J^T P &=& QR
\end{eqnarray*}
where $P\in O(m)$ is a permutation matrix, $Q\in O(n)$ and $R\in\reals^{n\times m}$ is an upper triangular matrix.
\begin{eqnarray*}
J &=& PLQ^T
\end{eqnarray*}
where $L = R^T$.
\begin{eqnarray*}
L &=& \left(\begin{array}{cc} L_1 & 0 \end{array}\right)\\
L^+ &=& \left(\begin{array}{c} L_1^-1 \\ 0 \end{array}\right)\\
L^+ L &=& \left(\begin{array}{cc}
I_r & 0 \\
0 & 0 \end{array}\right)
\end{eqnarray*}
where $r$ is the rank of $L$.
\begin{eqnarray*}
J^+ J &=& \left(\begin{array}{cc} Q_1 & Q_0\end{array}\right)
\left(\begin{array}{cc} I_r & 0 \\ 0 & 0\end{array}\right)
\left(\begin{array}{cc} Q_1 & Q_0\end{array}\right)^T \\
&=& Q_1 Q_1^T\\
I_n - J^+ J &=& Q_0 Q_0^T
\end{eqnarray*}
\end{document}
