I am interested in the tropical semiring (see Ref A for more details). For simplicity, in this case addition is redefined to $+ := \min$ and multiplication is redefined to be $\times := +$. Matrix multiplication remains defined as usual under the definition of matrix multiplication but uses the redefinitions of $+$ and $\times$.
Notice that using the usual technique of computing $\det(A - \lambda I)$ does not work for the tropical semiring as subtraction is not defined. In Ref A, a matrix (related to the adjacency matrix of a graph) is described on page 2 that I will refer to as the tropical adjacency matrix. The only eigenvector (see Theorem 3 of Ref A) is computed using a linear program (see Equation 7 of Ref A).
I am also interested in extending tropical semiring to the complex numbers. Of course, on one hand, it doesn't make sense to define the minimum complex number since the complex numbers cannot be made into an ordered field. However, if one plays fast and loose with $\min$, one sees that:
$$\min\{x, y\} = \frac{x + y + y\cdot\text{sign}(x - y) + x\cdot \text{sign}(y-x)}{2}$$
$\text{sign}$ can be extended to the complex numbers as $$\text{sign}(z) = \begin{cases} \frac{z}{|z|} & z \neq 0 \\ 0 + 0i & z = 0 \end{cases}$$ and therefore, $\min$ can be extended to the complex plane via complex addition, division, and the complex sign function. I'm not saying this is necessarily "the right" definition, but it was something I was curious about. I am interested in computing complex eigenvalues (in particular for tropical adjacency matrices) but I am unsure how to go about doing so (or even if any exist). Any guidance (even if it is just a nudge in the right direction) would be appreciated.
