Does anyone know bounds on the real parts of eigenvalues for generic real matrices? In particular I have a real matrix with complex eigenvalues with negative real parts, and I am looking for a bound on the maximum of the real parts (minimum absolute value) of the eigenvalues (to estimate the speed of convergence of a dynamical system).
In particular, is there any generalization of results like the following ones:
-For nonnegative irreducible matrices $A$ there is the following bound on the largest eigenvalue $\lambda_1$: $$ \min \sum_{j}A_{ij} \le\lambda_1\le\max \sum_{j}A_{ij} $$
-For real matrices and real eigenvalues, from Gershgorin circle theorem it follows that: $$|\lambda|\le \max \sum_{j}|A_{ij}| $$
In general I think that Bendixson's bound, which you mention in one of your comments, is the best one can do, without further constraints on $A$ or its antisymmetric part $K = (A - A^T)/2$.
The reason being that if $\lambda_1(S)$ is the largest eigenvalue of $S = (A + A^T)/2$, then $\text{Re}\{\lambda_1\}(A) \leq \lambda_1(S)$. If $A$ is normal, $A A^T = A^T A$, then it becomes an equality.
So, if it is not possible to compute $\lambda_1(S)$ explicitly, one option is to use bounds on the $\lambda_1(S)$, like Gershgorin circle theorem, or Laguerre/Samuelson bound, on $\lambda_1(S)$. However, they most likely will be less strict than Bendixson's bound.
Otherwise, you make $A$ traceless $\tilde{A} = A - \frac{\text{Tr}A}{n}I$ and look for bounds on its largest positive root $\lambda_1(\tilde{A})$ from the coefficients of the characteristic polynomial. Then, translate that bound back to $\lambda_1(A)$.
If the bound from the characteristic polynomial are tighter than Bendixson's bound, its because you have incorporated information about the antisymmetric part of $A$, when computing the characteristic polynomial.