What can we learn from the sign structure of the Jacobian matrix?

Question

I am studying a $4 \times 4$ Jacobian matrix. I know the sign structure (that is, I know whether each element is positive, negative or zero), but I do not know magnitudes of each elements (i.e. their numerical size). \begin{align} J = \begin{bmatrix} - & 0 & - & + \\[0.3em] + & + & - & + \\[0.3em] - & 0 & + & + \\[0.3em] 0 & + & 0 & 0 \end{bmatrix} \end{align}

I want to know what I can learn from the sign structure. For example, I know it is a saddle for a numerical example with the same structure.

Please offer pointers or directions to look in, especially regards stability analysis.

Answer 1

I think you might be looking for the Routh-Hurwitz stability criterion, which is closely related to the eponymous theorem. Basically, this relates the sign of subdeterminants of the matrix to the sign of the real parts of the eigenvalues of the original matrix -- which is quite relevant for stability analysis. For your specific structure, especially given the fact that quite a few entries are zero, you might get quite far using this approach.