Let $f(x) = -x^{T} A x$ for a positive definite matrix $A$. Let the domain of $x \in R^N$ be in a compact ball (with radius > $\sqrt{N}$ if necessary)
What is the Lipschitz constant for $f$ in terms of (presumably spectral) properties of $A$? Is it the maximum eigenvalue of $A$?
Any continuously differentiable function over a compact domain is Lipschitz continuous with Lipschitz constant equal to the maximum magnitude of the derivative.
The function $f(x) = x^TAx$ is differentiable with $\nabla f(x) = 2Ax$. It follows that over the ball of radius $R$, the Lipschitz constant of $f$ is equal to $$ \max_{x \in B_R} \|2Ax\| = 2R\cdot \max_{\|x\| \leq 1} \|Ax\| = 2R\cdot \rho(A), $$ where $\rho(A)$ denotes the spectral radius of $A$ (the maximal absolute value among the eigenvalues of $A$).