Routh-Hurwitz criterion for matrices 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
is very old and less specific. It gives no explicit inequalities, hence does not answer my question.)
 A: 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.
A: The very boring answer, of course, is:

*

*write down the characteristic polynomial $p(x) = \det(A-xI)$

*write down the Routh-Hurwitz criterion for $p$, expanding everything in terms of the matrix coefficients.

This is a scheme to generate them. I don't think there is a simple form for it in terms of the matrix coefficients for a general $n$, though.
For real $2\times 2$ matrices, I can tell you the explicit form because I used it very recently in a preprint:
\begin{align}
Tr(A) &\leq 0,\\
\det(A) &\geq 0
\end{align}
is a necessary and sufficient condition for the eigenvalues of a $2\times 2$ matrix to be in the (closed) left half-plane, and
\begin{align}
\det(A) &\leq 1,\\
Tr(A) &\leq 1 +\det(A),\\
-Tr(A) &\leq 1 +\det(A)\\
\end{align}
for the unit disc.
Just as a bonus, I find this picture particularly pretty: 
The cyan fat cross is the set of real $2\times 2$ matrices with eigenvalues in the unit disc and $A_{11}=A_{22}$, intersected with $[-2,2]^3$ (the four arms extend to infinity). The lines where two surfaces of the boundary cross correspond to matrices of the form $\begin{bmatrix}0 & a\\ 1/a & 0\end{bmatrix}$, $\pm \begin{bmatrix}1 & a\\ 0 & 1\end{bmatrix}$, $\pm \begin{bmatrix}1 & 0\\ a & 1\end{bmatrix}$.
