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 ) \leq 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.