Make capacitance a matrix with a reference

Fix the conductor geometry and assume a linear dielectric or vacuum between them, with no independently prescribed volume charge. Solve once with each electrode at unit potential and the others at zero. Superposition combines these basis potentials into any voltage configuration. Integrating the normal displacement around each conductor then gives its total charge as a linear combination of all conductor voltages. These coefficients are Maxwell capacitance coefficients, not arbitrary circuit fitting parameters.

A diagonal coefficient is positive and off-diagonal coefficients are nonpositive in the usual connected electrostatic setting. Raising one electrode’s voltage draws opposite-sign charge onto others held fixed. This sign convention differs from listing positive two-terminal capacitances in a circuit diagram, where the minus signs are written explicitly through voltage differences. State which convention is in use before comparing numerical tables. Reciprocity follows from Green’s identity, and exchanges the source and observation electrodes.

Include an enclosing reference conductor, or the appropriate infinity reference, as an additional index. Raising all potentials together changes no potential differences or fields. The complete matrix therefore annihilates the uniform vector, so every row sums to zero and its field-energy quadratic form has a gauge null direction. This statement includes the reference in the sum. A common voltage shift cannot be applied to the finite electrodes while secretly holding infinity at zero.

Once the reference is grounded, delete its row and column to form a reduced matrix. This matrix is normally positive definite and invertible for independent electrodes; its row sums need not vanish. They measure coupling to the removed reference. A floating electrode instead has fixed total charge and unknown voltage, which must be solved from its matrix row. Floating does not mean zero voltage, and a neutral floating electrode need not have zero local surface charge density.

For a passive linear medium, field energy is the integral of one half electric field dotted with displacement. It is nonnegative even when individual products of a conductor’s charge and voltage are negative. The total quadratic form, rather than each separate term, is the invariant physical quantity. Checking symmetry, signs, dimensions, and the relevant positivity can expose a table that cannot describe the proposed electrostatic geometry.

The matrix lab demonstrates why a positive determinant alone is a weaker statement. Reset and set diagonal scale to 1 and shear to 0; predict determinant 1 and condition number 1. Reset and set scale to 1 and shear to 1; predict the same determinant but unequal singular values and worse conditioning. These predictions concern readouts for the selected final matrix A. At Reset the square is still the identity; Play or Step interpolates geometry between identity and A. Its triangular matrix is nonsymmetric when shear is nonzero, so it is not a capacitance matrix or a solver for conductor charge.