CC BY 4.0 UnportedSimonson, L.Özdemir, S.K.Busch, K.El-Ganainy, R.2023-06-022023-06-022022https://oa.tib.eu/renate/handle/123456789/12305http://dx.doi.org/10.34657/11337The linear response of non-Hermitian resonant systems demonstrates various intriguing features such as the emergence of non-Lorentzian lineshapes. Recently, we have developed a systematic theory to understand the scattering lineshapes in such systems and, in doing so, established the connection with the input/output scattering channels. Here, we follow up on that work by presenting a different, more transparent derivation of the resolvent operator associated with a non-Hermitian system under general conditions and highlight the connection with the structure of the underlying eigenspace decomposition. Finally, we also present a simple solution to the problem of self-orthogonality associated with the left and right Jordan canonical vectors and show how the left basis can be constructed in a systematic fashion. Our work provides a unifying mathematical framework for studying non-Hermitian systems such as those implemented using dielectric cavities, metamaterials, and plasmonic resonators.enghttps://creativecommons.org/licenses/by/4.0620HermitiansInput-outputLinear responseLorentzian line shapeResolvent operatorsResonant systemsScattering channelsSystematic theoriesArticle