Okay, this is something that is bothering me. If $v$ is a vector on $\mathbb{R}^{n}$ and we want to project it onto some subspace, say, $V \subseteq \mathbb{R}^{n}$, we look for a projection matrix $P$ and the projection is given by $Pv$.
Is it possible to project matrices as well? If so, suppose I want to project a matrix $M$ onto some vector subspace of $\mathbb{R}^{n}$. Is this projection given by $PM$ or $PMP$? I realized I'm getting confused about this.
As a direct application of what I'm asking; let $M$ and $N$ two matrices. Suppose $P$ projects $N$ to zero. Should this imply $PNP = 0$ or simply $PN = 0$? In this case, is $PM$ (or, $PMP$) the projection of $M$ onto the same subspace?