Given a matrix $A \in \mathbb{R}^{m \times n}$ and denotes $C := \left\lbrace \left( x , z \right) \in \mathbb{R}^{n} \times \mathbb{R}^{m} \colon A x - z = 0 \right\rbrace$.
Is there any way to construct a projection map onto C?
In particular, for any $\left( u , v \right) \in \mathbb{R}^{n} \times \mathbb{R}^{m}$, how to find the point $\Pi_{C} \left( u , v \right) := \left( \Pi_{C}u , \Pi_{C}v \right) \mathbb{R}^{n} \times \mathbb{R}^{m}$ such that $A \left( \Pi_{C}u \right) - \Pi_{C}v = 0$?
If we regard $(x,z)$ as a vector in $\Bbb R^{n+m}$, then $C$ is simply the column space of the matrix $$ M = \pmatrix{I\\A} $$ The matrix of the projection onto this column space may be computed as $$ M[M^TM]^{-1}M^T = M[I + A^TA]^{-1}M^T $$ That is, given a $(u,v)$, the projection is given by $$ \pmatrix{u_{proj}\\v_{proj}} = M[M^TM]^{-1}M^T\pmatrix{u\\v} $$