It is well-known with respect to Routh-Hurwitz Criterion that for an arbitrary matrix $A$ with real coefficients, one can derive a series of analytic expressions with these real coefficients, so as to determine whether all real parts of eigenvalues for the matrix $A$ are negative, without solving the corresponding polynomial. However, if there exist some complex coefficients in the matrix $A$, Routh-Hurwitz Criterion may be not feasible to deal with the aforesaid issue.
In my research work, I have derived a matrix as follows: \begin{align} X = \left( \begin{array}{ccccc} {0} & 1 & 0 & \cdots & {0} \\ 0 & 0 & \ddots & \ddots & \vdots \\ \vdots & \vdots & \cdots & 1 & 0 \\ 0 & 0 & \cdots & \cdots & 1 \\ -g_0 \lambda & -g_1 \lambda & \cdots & \cdots & -g_{m-1} \lambda \end{array} \right) \text{.} \end{align}
where $g_i > 0$, $i = 0, 1, …, m - 1$, $m$ is a positive integer with $m > 1$, and $\lambda$ is a complex number. I would like to derive a criterion, similar to Routh-Hurwitz Criterion, in order to determine whether all real parts of eigenvalues for the matrix $X$ are negative via the coefficients of $X$, instead of solving the corresponding polynomial. Unfortunately, I have never found an available sufficient condition yet. Here, I am looking for your help sincerely, and thanks for your kind attention.
