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.)