# Why does the RG group flow's linearization provide an eigenbasis at fixed points?

I'm reading Conformal Field Theory by David Sénéchal, Philippe Di Francesco, and Pierre Mathieu.

Let $T$ be the map that generates the renormalization (semi-)group by taking couplings $J$ to $J'$ (these are vectors whose entries are coupling constants),

$$J' = T(J).$$

Let $J_c$ be a fixed point of $T$. They claim that at $J_c$ the linearization of $T$ is diagonalizable and provides an eigenbasis for the space of coupling constant vectors. I do not see why this is; is my multivariable calc just failing me or is there a physical reason to expect this? Thanks!