Prove a $C^{\infty}$ multivariable function is lipchitz via Jacobian matrix I would like to prove a real $C^{\infty}$(polynomial) multivariable function $F : (a_1,a_2,...a_n) \rightarrow (b_1,b_2,...b_n) $ is lipchitz of parameter $l$
is it sufficient to prove the norm of dominant eigenvalue of $Jacobian(F)$ is less than $l$.
In other words, is the "spectral radius of Jaobian matrix is less than 1" a sufficient condition for $F$ to be lipschitz of paramter $l$?
 A: As Nate's example shows, the eigenvalues aren't enough, but the Lipschitz constant is bounded by the square root of the largest eigenvalue of $J^*J$ or $J J^*$.  This is the largest singular value of the Jacobian.  The singular value decomposition decomposes $J$ as $UDV$, where $U$ and $V$ are orthogonal matrices and $D$ is a diagonal matrix whose entries are the singular values of $J$, so the largest singular value is the length of the major axis of the image of the unit ball under $J$.
A: It looks to me that $F$ is Lipschitz iff  each of the components $b_1,\dotsc, b_n$ is Lipschitz. If one of the components, say $b_1$,  has degree $\geq 2$ it cannot be Lipschitz.  Indeed, in this case $\newcommand{\bR}{\mathbb{R}}$
$$\sup_{a\in\bR^n}|\nabla b_1(a)|=\infty. $$ 
Given a  constant $L>1$ choose  $a\in \bR^n$ such that $|\nabla b_1(a)|>L$. Set $h:=\nabla b_1(a)$. Then
$$\lim_{t\to 0} \frac{1}{t}(b_1(a+th)-b_1(a)= |h|^2>L|h| $$
so that for $t>0$ sufficiently small
$$\frac{1}{t|h|} |b_1(a+th)-b_1(a)|> L.$$
