I was wondering if there is a closed formular for the projection onto the intersection of the subspaces $Ax = b$ and $Zx = 0$. I know there is a closed formula for either one of those, but can you also project onto the interesection by use of the pseudoinverse?
I am aware of the alternating projection method, but this takes too long for my purposes.
Thanks!
The problem is given by:
$$\begin{aligned} \arg \min_{x} \quad & \frac{1}{2} {\left\| x - z \right\|}_{2}^{2} \\ \text{subject to} \quad & A x = b \\ & Z x = 0 \end{aligned}$$
Which is equivalent of:
$$\begin{aligned} \arg \min_{x} \quad & \frac{1}{2} {\left\| x - z \right\|}_{2}^{2} \\ \text{subject to} \quad & C x = d \\ \end{aligned}$$
Where $ C = \begin{bmatrix} A \\ Z \end{bmatrix} $ and $ d = \begin{bmatrix} b \\ 0 \end{bmatrix} $.
Then the answer is given by $ x = z - {C}^{T} {\left( C {C}^{T} \right)}^{-1} \left( C z - d \right) $.
It can be easily generalized to more than 2 sub spaces.