Max-plus definite matrix closures and their eigenspaces
This paper provides a contribution to the geometrical understanding of earlier algebraic results on max-plus eigenspaces [cf. \textit{P. Butkovič}, Discrete Appl. Math. 105, 73--86 (2000; Zbl 0976.15013); \textit{P. Butkovič} and \textit{R. A. Cuninghame-Green}, Linear Algebra Appl. 421, No. 2--3, 370--381 (2007; Zbl 1131.15008)] by regarding the max-plus semiring \(\mathbb R _{\max}:= \mathbb R \cup \{-\infty\}\) with its simplifying total order and using basic tools of max-algebra. Definite forms for a given matrix \(A= (a_{ij}) \in \mathbb R^{n\times n}_{\max}\) with nonzero (finite) permanent are considered and it is shown that the closures of all of them coincide, so resulting in the ``definite closure'' operation. The author presents a description of the eigenspace \(eig(A)\) and the faces of it for a definite matrix \(A\). Employing a representation (op. cit.), he concludes that \(eig(A)\) has a non-empty interior if and only if \(A\) has a strong permanent. Finally, this interior is fully detailed by means of Hilbert distances.
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