When is “$\Re(\lambda) \gt 0$ for $\lambda \in \sigma(A),A \in \mathbb{R}^n $” true?

Question

Let $A \in \mathbb{R}^{n \times n}$ and $\sigma(A)$ the spectrum of $A$. I am searching for a fast way to check whether $\Re(\lambda) \gt 0$ for all $\lambda \in A$.

If $A = A^t$, one only has to check whether all leading principal minors are positive. If one of these is negative, there has to be a negative eigenvalue and if one of these is zero, then the test does not give any information.

Does there exist a similar (fast) approach for non symmetric matrices?

Answer 1

Firstly , you calculate $p$, the characteristic polynomial of $-A$. The question is: is $p$ a Hurwitz polynomial ? The answer is given by the Routh-Hurwitz stability criterion

http://en.wikipedia.org/wiki/Routh%E2%80%93Hurwitz_stability_criterion

This is a tabular method; you can do that with hand or using the Maple library

DynamicSystems[RouthTable]