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The "full" (your term) capacitance matrix Cij which you are defining
restricts the validity of the equation Qi=SUM CijVj to states with the
same total charge, SUM Qi.
There is no one set of Cij that is a property
only of geometry and the choice of voltage reference point.
This is not very useful.
If I have an isolated system of two
separated conductors, I would like an equation Qi=SUM Cij Vj which defines
the Cij as properties of the system geometry (and the gauge) and are
independent of the total system charge, so that I can consider cases in
which Q1 = -Q2, or Q1=2*Q2, or Q1 = -3*Q2, or etc, using the same set of
Cij. I want to use the same Cij to describe all various ways of
depositing charges on these conductors, holding fixed only the geometry
and the voltage reference space-point.
Adding the constraint of fixed total charge
is doing a very different, and severely restricted, problem.