  In this paper we perform the exact diagonalization of a light-matter strongly
coupled system taking into account arbitrary losses via both energy dissipation
in the optically active material and photon escape out of the resonator. This
allows to naturally treat the cases of couplings with structured reservoirs,
which can strongly impact the polaritonic response via frequency-dependent
losses or discrete-to-continuum strong coupling. We discuss the emergent gauge
freedom of the resulting theory and provide analytical expressions for all the
gauge-invariant observables both in the Power-Zienau-Woolley and the Coulomb
representations. In order to exemplify the results the theory is finally
specialised to two specific cases. In the first one both light and matter
resonances are characterised by Lorentzian linewidths, and in the second one a
fixed absorption band is also present. The analytical expressions provided in
this paper can be used to predict, fit, and interpret results from polaritonic
experiments with arbitrary values of the light-matter coupling and with losses
of arbitrary intensity and spectral shape, in both the light and matter
channels.
