Is the stability matrix of a linearised RG flow always diagonalisable? This is a follow up on "Why are the eigenvalues of a linearized RG transformation real?".
My question is simple: Is there some physical (or mathematical) reason for the stability matrix of Renormalisation Group flows close to fixed points to be diagnoalisable? What is it? If there isn't: Are there known counter examples? How do we deal with them?
 A: I don't believe there is a mathematical reason, especially if there is latitude in reverse-engineering the field theory or stat mech system to evince such a behavior. Indeed, if Lorentz-nonivariant systems are examined, things like limit cycles , e.g. this one are not hard to concoct. As for physical reasons, they might well be easy to bypass/moot if one argued for them. I don't know of any systems, however, with this property, which might not say much. 
As a mathematical wisecrack, I could manufacture a simple toy system with two couplings, x and y and logarithmic scale variable t :
$$
\dot{x}=-x + ay, \qquad \dot{y}= -y , 
$$
with evident solutions stable around the fixed point (0,0),
$$
y= e^{-t}, \qquad x= (c +at) e^{-t} .
$$
The stability matrix of the ODE system is
$$
\left( \begin{array}{cc}
-1 & a  \\
0 & -1 \\
\end{array} \right)
$$
which is not diagonalizable, with only one eigenvector,
$$
\left( \begin{array}{c}
1 \\
 0 \\
\end{array} \right)
$$
of eigenvalue -1.  This is not to say the system is not stable, however, if one could solve the ODE, somehow, as here.
