derivative of a positive definite matrix

Question

Suppose that $A$ is a positive definite symmetric matrix, specifically a Riemannian metric. Can we say anything about the sign of $tr(A^{-1}\partial_i A)$?

Answer 1

Assuming $\partial_i$ comes from a coordinate frame (so that $\nabla_i \partial_j = \nabla_j \partial_i$), your expression is

$$ {\rm tr}(g^{-1} \partial_i g) = g^{jk}(g(\nabla_i \partial_j, \partial_k)+g(\partial_j,\nabla_i \partial_k))=2\Gamma^k_{ik}=2\textrm{ div }\partial_i.$$

From a geometric point of view this is the volume expansion as you flow along the vector field $\partial_i$; so a priori you certainly can't conclude anything about its sign. (You should be able to think of some expanding and contracting flows that can (at least locally) form one axis of a coordinate system.)

This interpretation is made precise by the equalities

$$ \mathcal L_X d\mu = \mathrm{div}(X)\ d\mu = \frac12 {\rm tr}_g \mathcal L_Xg;$$

so assuming this isn't where your question originated you should at least have some more intuition about it now.