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-singular matrix with a non-degenerate null eigenvalue. We explore the physical implications of this fact, and study the physical meaning of the eigenvalue problem for such a matrix. Many properties are easily visualized by constructing a “potential space” isomorphic to the euclidean space. The problem of minimizing the internal energy of a system of conductors under constraints is considered, and an equivalent capacitance for an arbitrary number of conductors is obtained. Moreover, some properties of systems of conductors in successive embedding are examined. Finally, we discuss some issues concerning the gauge invariance of the formulation.

► The matrix of capacitance is a real positive and singular matrix.
► The gauge invariance requires the existence of a null non-degenerate eigenvalue of this matrix.
► A given eigenvalue is proportional to the internal energy associated with the eigenvector.
► Eigenvectors and eigenvalues can be measured, and give information about the matrix of capacitance.
► The reciprocity theorem is a consequence of the hermitian nature of the matrix.
► An equivalent capacitance for an arbitrary configuration of capacitors can be obtained.
► For successively embedded conductors, many elements of the matrix of capacitance are null.