Given an affine set $Ax=b$, the Projection operator to this set is $$P(z) = z - A^{T}(AA^{T})^{-1}(Az-b)$$ which is also affine.
How is this derived?
2026-04-02 01:32:11.1775093531
Deriving projection operator for an affine set
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$P(z)$ solves the problem $$ \min \|z -x\|^2 $$ subject to the constraint $$ Ax=b. $$ The KKT system is a necessary and sufficient optimality condition (why?): $$ Ax = b, \ A^T\mu = x-z. $$ Multiply the second equation by $A$, assume $(AA^T)^{-1}$ being invertible, then solve for $\mu$, then solve for $z$.