$Ax=b$ is a linear system of equations in which $A$ is a square invertible matrix. However, I would like to approximately solve $Ax=b$ in which the constraint $x=Cy+d$ is exactly enforced. $C$ is $m\times n$ where $m>n$.
I need analytical solutions and am not interested in numerical approaches.
Thank you
Well, assuming that "approximately solve" and "analytic" (and that the matrices are solved so thte equation makes sense) I guess you want to minimize the squared norm of the error:
$$A(Cy+d)-b,$$
over $y$.
That is, you want to minimize the squared norm of
$$(W y - V),$$
with $W \equiv AC, V \equiv b-A d$.
To do that you would need to satisfy the first-order conditions, so I'll differentiate the squared norm of the above expression with respect to the vector $y$ to yield
$$W^T W = W^T V.$$
From there, you'd need to have an analytic technique to solve the linear equation, which might not be practical for large matrices. I think it works out pretty much the same for complex matrices, substituting conjugate transposes for transposes.