From Wiggins' book (introduction to applied nonlinear dynamical systems and chaos) below thing I am not follwing ..
"If all eigenvalues of a Jacobian matrix $J_{n\times n}$ of a vector valued function have negative real part then there exists a basis such that $(x.Jx)<k|x|^2<0$ for some real number $k$ and for all $x \in R^n$.
But here A real matrix whose eigenvalues have all negative real parts is a counter example.
Could you please help me to understand this?
Best Answer
I haven't access to this book. So, I don't know the context and I can only guess. Please ignore this answer if my guess is wrong.
I think the author is talking about a case where you change the basis first, then take the Jacobian matrix $J$ and study the inner product $(x,Jx)$. So, the change of basis occurs only to $J$ but not to $x$. In other words, what he/she actually meant may be the following:
Proposition. Suppose all complex eigenvalues of $J\in M_n(\mathbb R)$ have negative real parts. Then there exists a real number $k$ and a real invertible matrix $P$ such that $(x,P^{-1}JPx)<k|x|^2<0$ for all $x\in\mathbb R^n$.
Note that this does not contradict Robert Israel's counterexample in the thread you have mentioned, because our $P$ here is not necessarily real orthogonal. The proof of the above proposition is pretty standard: every real matrix can be block-triangularised to the real Schur form. If one carries out a further similarity transform via a suitable block-diagonal matrix, one can diminish the off-(block-)diagonal entries at will, so that the resulting matrix is arbitrarily close to a block diagonal matrix, each of whose diagonal blocks is either a $2\times2$ rotation matrix with a negative real part or a $1\times1$ negative scalar. The symmetric part of the resulting matrix is therefore negative definite.