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Max-plus definite matrix closures and their eigenspaces. (English) Zbl 1131.15009
This paper provides a contribution to the geometrical understanding of earlier algebraic results on max-plus eigenspaces [cf. P. Butkovič, Discrete Appl. Math. 105, 73–86 (2000; Zbl 0976.15013); P. Butkovič and 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.

MSC:
15A18 Eigenvalues, singular values, and eigenvectors
06F15 Ordered groups
15A30 Algebraic systems of matrices
15B33 Matrices over special rings (quaternions, finite fields, etc.)
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