The Routh-Hurwitz criterion explicitly specifies a finite set of inequalities on the coefficients of a polynomial, necessary and sufficient that all zeros lie in the unit circle or in the left half complex plane.


Is there a similar set of explicit inequalities on the coefficients of a (real or complex) matrix, necessary and sufficient that all eigenvalues lie in the unit circle or in the left half complex plane? I am not looking for a decision algorithm (one could just compute the eigenvalues...). Instead, I'd like to have explicit inequalities for 2 x 2 and 3 x 3 matrices (with an elegant proof), and a scheme to generate the inequalities for larger matrices.


(The related page
[Routh-Hurwitz for eigenvalues](https://mathoverflow.net/questions/58701/routh-hurwitz-for-eigenvalues)
is very old and less specific. It gives no explicit inequalities, hence does not answer my question.)



Routh-Hurwitz criterion for matrices
Hmm, a "similar set". Is the following similar enough? I do not know. But it has to be pointed out that there are of course very nice and largely tractable numerical tests for stability of matrices (and much better than attempting to compute the eigenvalues). Let $I$ denote the identity matrix. The spectrum of a (real or complex) matrix $A$ is in the left half plane, if and only if there is a positive definite, Hermitian $P$ such that
$$A^\* P + PA = -I.$$
For the unit circle the respective equation is
$$ A^\*PA - P = -I.$$
These are the so-called Lyapunov equations. The answer is a bit tongue in cheek, but still. First you have a bunch of linear equalities for the entries of $P$ using the coefficients in $A$. Then you need to check positive definiteness. To do this numerically, as an example Cholesky factorization has been [proposed](https://www.sciencedirect.com/science/article/pii/0024379588902236).


