电容矩阵的正性和其它性质：物理和数学含义
The positivity and other properties of the matrix of capacitance: physical and mathematical implications
摘要：
证明了静电学中的电容矩阵是具有非退化零特征值的正奇异矩阵。我们探讨了这一事实的物理含义，并研究了这样一个矩阵的特征值问题的物理意义。通过构造一个同构于欧几里得空间的“势空间”，许多性质很容易被形象化。考虑了约束条件下导体系统的内能最小化问题，得到了任意导体数的等效电容。此外，还研究了连续嵌入导体系统的一些性质。最后，我们讨论了有关规范不变性的一些问题。
英文摘要：
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.
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