Sufficient conditions for symmetric ($m \times m$) block matrices to be negative definite if diagonal blocks are negative definite

Question

Consider a symmetric block matrix $$A = \begin{bmatrix} A_{11} & \dots & A_{1m}\\ \vdots & \ddots & \vdots\\ A_{m1} & \dots & A_{mm} \end{bmatrix}~,$$ where each diagonal block $A_{ii}$ is negative definite. We can assume we know all entries of each block of $A$ or function(s) that will give you the entries (e.g., in cases where A is a Jacobian).

I would like to know sufficient conditions on the off-diagonal blocks that ensure $A$ as a whole is negative definite. Several relatively well known results exist for $A$ with rather special structures (e.g., diagonal, tri-diagonal, and Toeplitz). Also, several other posts on here seem to focus specifically on matrices partitioned into $2 \times 2$ blocks. However, I was hoping to find some more general characterizations when negative definiteness of the whole is preserved.

Thus far I have found the following ways of testing the definiteness of block matrices in general:

• Block diagonal generalizations of Gershgorin-type theorems. (For example, those found in citation 1 below and the references within.)
• Characterizations of the spectrum of $A$ based on the traces of $A$ and $A^2$. (Corollary 2.2 in citation 2 below.)

Any general characterizations of the definiteness of block matrices is appreciated since I would love for this post to be maximally useful for folks looking into similar problems in the future. That said, I am specifically working with block matrices arising from the Jacobian of a system of ODEs. The problem I am considering is as follows: We have a system of $m$ agents where each agent has a strictly concave utility function on some convex domain of parameters. The ODEs are the gradients of each agent's utility function, the diagonal blocks of $A$ correspond to the Hessian of each agent's utility function, and the off-diagonal blocks correspond to higher-order interactions between agents. The negative definiteness of diagonal blocks follows from the strict concavity of the utilities, and I am trying to derive conditions on utility functions so that fixed points of this system are stable.

Thanks in advance!

Citations:

1. J. Kierzkowski, and A. Smoktunowicz. Block normal matrices and Gershgorin-type discs
2. H. Wolkowicz, and G. P. H. Styan. Bounds for eigenvalues using traces