I don't know if this is the answer that you have been looking for, but let me offer a rather trivial observation.

Your tensor is the Lie derivative of the metric tensor with respect to the vector field $\textbf u$. Loosely speaking, the Lie derivative $\mathcal{L}_\textbf{u}$ has an interpretation as a derivative with respect to "dragging" a tensor along the flow defined by $\textbf{u}$. A simple manipulation shows that for any vector field $\textbf{v}$,
\begin{equation}
\mathcal{L}_\textbf{u} (\textbf{v} \cdot \textbf{v}) = 
2 \textbf{v} \cdot \mathcal{L}_\textbf{u} \textbf{v} - 2 \textbf{v} A \textbf{v} \,,
\end{equation}
where the matrix $A$ is the one you have defined. A flow $\textbf{u}$ with positive definite $(-A)$ has the property that
\begin{equation}
\mathcal{L}_\textbf{u} (| \textbf{v} |^2 ) \geq 
2 \textbf{v} \cdot \mathcal{L}_\textbf{u} \textbf{v}
\end{equation}
for any vector field $\textbf{v}$.

I hope this observation has some use to you.