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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.

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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