Representation of the Orthogonal Projection We want to show that orthogonal projection can be written as
\begin{align}
C\cdot C^+=\left(I_N-\frac{1}{N}\iota\iota'\right),
\end{align}
where $C=\left(\omega\iota'-I_N\right)$, $I_N$ is the identity matrix of dimension $N$, $\omega$ is an $N\times1$ vector, $\iota'\omega=1$ and $C^+$ denotes the Moore–Penrose pseudo-inverse of $C$. Numerically, we know that this holds for any vector $\omega$ fulfilling the summing-up-constraint. 
Here, $C$ is a linear operator from $\mathbb{R}^{T\times N}$ onto $\mathbb{R}^{T\times N}$ and we want to show that despite of the choice of $\omega$, the orthogonal projector $CC^+$ is always of the same form.

We find finally that $C^+$ has the following form:
\begin{align}
C^{+}= w \iota' - I  \frac{1}{\omega'\omega} \omega \omega' - \left(  \frac{1}{N} \frac{1}{\omega'\omega} + 1\right) \omega \iota' + \frac{1}{N} \iota \iota'
\end{align}
