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The secular evolution of an infinitely thin tepid isolated galactic disc madeof a finite number of particles is investigated using the inhomogeneousBalescu-Lenard equation expressed in terms of angle-action variables. Thematrix method is implemented numerically in order to model the inducedgravitational polarization. Special care is taken to account for theamplification of potential fluctuations of mutually resonant orbits and theunwinding of the induced swing amplified transients. Quantitative comparisonswith ${N-}$body simulations yield consistent scalings with the number ofparticles and with the self-gravity of the disc: the fewer particles and thecolder the disc, the faster the secular evolution. Secular evolution is drivenby resonances, but does not depend on the initial phases of the disc. For aMestel disc with ${Q \sim 1.5}$, the polarization cloud around each star boostsup its secular effect by a factor of the order of a thousand or more, promotingaccordingly the dynamical relevance of self-induced collisional secularevolution. The position and shape of the induced resonant ridge are found to bein very good agreement with the prediction of the Balescu-Lenard equation,which scales with the square of the susceptibility of the disc.  In astrophysics, the inhomogeneous Balescu-Lenard equation may describe thesecular diffusion of giant molecular clouds in galactic discs, the secularmigration and segregation of planetesimals in proto-planetary discs, or eventhe long-term evolution of population of stars within the Galactic centre. Itcould be used as a valuable check of the accuracy of ${N-}$body integratorsover secular timescales.

Secular diffusion in discrete self-gravitating tepid discs II: accounting for swing amplification via the matrix method