Methods to solve system of linear equations with equiality constraint

Hello, everyone. I want to solve a system of linear equations

Ax = b

with some linear equality constraints. A is square but non-symmetric.
The equation corresponds to a linearized stationary Boltzman equation, and the equality constraints correspond to the boundary conditions and restrict the angular harmonics of the distribution.

My question is, what is the go to method to solve the system of linear equations with linear equality constraints? Of course, one could define the system of coordinates where the constraints are taken into account, however, in my case it is inconvenient.

I usually use Lagrange multipliers; this just involves solving an augmented system which includes some new unknowns (Lagrange multipliers) which help enforce the constraint.

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Specifically, if your constraints are Cx=f, the system would be

[A C’][x ] = [b]
[C O][lambda] [f]

Does this approach works well when A is neither symmetric nor Hermitian?

I’ve used it before in that setting, but I’m not sure if it always works “well”. IIRC the system is derived for SPD matrices A in the context of optimization; if A is not SPD then this interpretation doesn’t hold, but I think it still “works”.

If the constraint is consistent with the system I think it generally works OK. Either way, if you have access to the matrix, it’s not hard to set up the system and try it out.