Tropical compound matrix identities

Marianne Akian (1, 2) , Stéphane Gaubert (2, 1) , Adi Niv (3)

1 CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique
2 TROPICAL - TROPICAL
3 Shamoon Academic College of Engineering [Beer-Sheva]

We prove identities on compound matrices in extended tropical semirings. Such identities include analogues to properties of conjugate matrices, powers of matrices and $\operatorname{adj}(A)\det(A)^{ -1}$, all of which have implications on the eigenvalues of the corresponding matrices. A tropical Sylvester-Franke identity is provided as well.

Mots clés: Pseudo-inverse; Permanent; Eigenvalues; Tropical linear algebra; Definite matrices; Compound matrix; Characteristic polynomial

Domaines: Optimisation et contrôle [math.OC]

Marianne Akian, Stéphane Gaubert, Adi Niv. Tropical compound matrix identities. Linear Algebra and its Applications, 2018, 551, pp.162-206. ⟨10.1016/j.laa.2018.04.006⟩. ⟨hal-01469638⟩