Theory of Characteristic Modes for Nonsymmetric Surface Integral Operators - Yla-Oijala, Pasi; Wallen, Henrik (IEEE transactions on antennas and propagation, 2021)
The theory of characteristic modes is formulated with nonsymmetric surface integral operators for perfect electric conductors, impedance surfaces, and homogeneous dielectric bodies. For nonsymmetric (nonself-adjoint) operators, the eigenvectors are not orthogonal with respect to the weighted inner product defined with the weighting operator of the generalized eigenvalue equation. Rather, this orthogonality holds between the eigenvectors of the original equation and the adjoint equation, including adjoint operators. This implies that the modal expansion, used to express any scattering or radiation solution as a linear combination of the modes, requires these two sets of eigenvectors. For matrix equations, the eigenvectors of the adjoint equation correspond to the left eigenvectors of the original equation.