Stability of a matrix with complex coefficients

Question

while solving a physical problem of an optical beam propagating through a periodic media, I have obtained the following system of coupled differential equations

\begin{gather} \frac{d}{dz}\begin{bmatrix} y_1 \\ y_2 \\ y_3 \end{bmatrix} = \begin{bmatrix} V_{11} & V_{12} & V_{13} &\\ V_{21} & V_{22} & V_{23} &\\ V_{31} & V_{32} & V_{33} &\\ \end{bmatrix} \begin{bmatrix} y_1 \\ y_2 \\ y_3 \end{bmatrix}, \label{matrix1} \end{gather} where all of the $V_{ij}$'s are complex valued. My question is: how can one study the stability of a system represented by a general complex $n\times n$ matrix? I've only found one specific paper on Google about stability in matrices with complex coefficients [Z. Zahreddine, E. F. Elshehawey, "On the stability of a system of differential equations with complex coefficients" Indian J. Pure Appl. Math. 19, 963 (1988)] but it is still not clear how to proceed with the above matrix and the paper has only 61 citations according to Google Scholar but none of the others papers has helped me.

Are there any other mathematical resources on stability of matrices with complex coefficients?

I really appreciate all the help!

Answer 1

There's nothing complicated about it. As AcccidentalFourierTransform says in a comment:

"You just replace [the criterion for stability] $\lambda_i < 0$ with $\mathrm{Re}(\lambda_i) < 0$ [for each and every $i$]"

and this holds for exactly the same reason as holds for in the real valued case. Namely, reduce the matrix to Jordan normal form, so there exists a system of co-ordinates $\tilde{y}_i$ related to then original by a similarity such that that we get decoupled equations of the form:

$$\frac{\mathrm{d}}{\mathrm{d}z} \tilde{y}_i= (\lambda_i\,\mathrm{id} + U_i)\,\tilde{y}_i$$

where $U_i$ are upper triagular and idempotent with noughts along the diagonal and ones along the superdiagonal and noughts elsewhere. The solutions then contain only growth factors of the form $\exp(\lambda_i\,t)$, so that these remain strictly dwindling with increasing $z$ iff $\mathrm{Re}(\lambda_i)<0$ and bounded with increasing $z$ iff $\mathrm{Re}(\lambda_i)\leq0$

Answer 2

At 2017 American Control Conference last year in Seattle I attended a talk that addressed systems in control that involve complex coefficient models in transfer function and state space forms. A paper was published in the conference proceedings entitled Complex-Coefficient Systems in Control.

The paper does address the question of stability from various modeling methods and offers further references as well. I asked the speaker what specific characteristic leads to complex-coefficient systems. I recall he couldn't answer definitively and after reading the paper it just offered examples rather than generalizing the question. So it appears it's a ripe area of study. From what I've seen all the examples involve some form of nonlinearity or dimensionality greater than one. I suppose you can add optical systems to the list? Modulation systems seem to be tractable by such models.

The paper does have a section on State Space Analysis which is close to what you are trying to analyze. It says that the concepts of controlability and observability extend to complex coefficient systems and the Bode integral applies for transfer function models, however you have to integrate negative frequency as well as positive.

Hope that helps.