In this paper, exactly at page 11, I find the following inequality:
$\lVert A - CC^\dagger A \rVert \leq \sqrt{1 + k(n-k)} \cdot \lVert A - A_{(k)} \rVert $
Where $A$ is an $m\times n$ matrix, $C$ is a $k$-column submatrix of $A$, and $A_{(k)}$ is the best rank-$k$ approximation of A. So that inequality is basically a statement about the properties of a matrix $C$ obtained by picking $k$ columns from $A$.
Now the question is: what does $CC^\dagger A$ represent exactly? By looking around I found that it could be a kind of projection of matrix $A$, but I really don't get the point. I also don't understand where the expression $CC^\dagger A$ comes from, and how it can be derived.
I would really appreciate your help in understanding this