Summary: Appendix 1 ­ Stability
We calculate the Jacobian matrix Dfw, for a fixed vector w:
Lemma 0.1. Dfw = I + C - 2w(Cw)T - (wT Cw)I
Proof. Call g(w) = (wT Cw)w , so f(w) = w + (Cw - g(w))
gi(w) = (wT
Cw)wi
If i = j:
gi
wj
(w) =

wj
(
k,l
Cklwkwl)wi = 2(
k
Ckjwk)wi = 2[Cw]jwi
If i = j:
gi
wi

Source: Adams, Paul R. - Department of Neurobiology and Behavior, SUNY at Stony Brook

Collections: Biology and Medicine