The positivity and other properties of the matrix of capacitance: physical and mathematical implications

We prove that the matrix of capacitance in electrostatics is a positive matrix. We explore the physical implications of this mathematical fact, and study the eigenvalue problem for the matrix of capacitance and its physical meaning. Many properties are easily visualized by constructing a “potential space” isomorphic to the euclidian space and defining an inner product in it. The problem of minimizing the internal energy of a system of conductors under constraints is considered, and an equivalent capacitance is obtained for an arbitrary number of conductors. Moreover, some properties of systems of conductors in succesive embedding are examined. Finally, we prove that the formulation utilized here is gauge invariant while the one used in the literature is not.