= Damping of coupled secular eigenmodes
{title2=$A'=A+i\operatorname{diag}(\tau^{-1},0)$}
Linear <eccentricity damping> on one planet changes $\dot z=iAz$ to $\dot z=i[A+i\operatorname{diag}(\tau^{-1},0)]z$. For a nondegenerate, weakly damped two-planet matrix, let $D=\sqrt{(A_{11}-A_{22})^2+4A_{12}A_{21}}$. The modal damping rates are $g_{p,m}=[1\pm(A_{11}-A_{22})/D]/(2\tau)>0$. Both <secular eigenmodes> damp even though only one planet is directly affected; the coupling shares the dissipation.
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