Automated diagonalization of Maxwells equations: Theory, implications and applications

Abstract

The 3 × 3 identity matrix is additively and multi-plicatively factorized into unitary scaffolding matrices, using the Frobian-Schmidt matrix norm. The decomposition, being ambiguous, leads to various realizations of scaffolding matrices, corresponding to pairs of operators and their dual counterparts in mathematical physics. The scaffolding matrices neatly capture the property that the negative gradient and divergence operators are adjoint. It also accounts for self-adjointness of the curl operator. The scaffolding matrices are used to diagonalize Maxwell's equations in physically-realizable, fully-bianisotropic inhomogeneous media. It is shown that the diagonalization process can be automated, following a sequence of algorithmically smooth operations. The existence of such a recipe is the gist of the paper. In the absence of any impressed sources, diagonalized forms transform to equivalent eigenvalue equations in spectral domain. A further major result is that the interface conditions are implicit in the theory; they arise from the formulation automatically, without resorting to the text-book approach of introducing a pill-box, and performing a limiting process. A plethora of theoretical implications and practical recipes follow from the developed theory, attesting to the unifying and fundamental character of diagonalization. A list of unsolved challenging problems is presented, including the question as to why several other possible realizations of scaffolding matrices do not play any role in mathematical physics.