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Spectral inequalities for nonnegative tensors and their tropical analogues

Shmuel Friedland 1 Stéphane Gaubert 2, 3 
CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique, Inria Saclay - Ile de France
Abstract : We extend some characterizations and inequalities for the eigenvalues of nonnegative matrices, such as Donsker–Varadhan, Friedland–Karlin, Karlin–Ost inequalities, to nonnegative tensors. Our approach involves a correspondence between nonnegative tensors, ergodic control and entropy maximization: we show in particular that the logarithm of the spectral radius of a tensor is given by en entropy maximization problem over a space of occupation measures. We study in particular the tropical analogue of the spectral radius, that we characterize as a limit of the classical spectral radius, and we give an explicit combinatorial formula for this tropical spectral radius.
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Submitted on : Thursday, January 3, 2019 - 3:43:44 PM
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Shmuel Friedland, Stéphane Gaubert. Spectral inequalities for nonnegative tensors and their tropical analogues. Vietnam Journal of Mathematics, Springer, 2020, Special Issue in the Honor of Volker Mehrmann, 48 (4), pp.893-928. ⟨10.1007/s10013-020-00432-0⟩. ⟨hal-01969021⟩



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