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Journal Articles Vietnam Journal of Mathematics Year : 2020

Spectral inequalities for nonnegative tensors and their tropical analogues

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.

Dates and versions

hal-01969021 , version 1 (03-01-2019)

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Shmuel Friedland, Stéphane Gaubert. Spectral inequalities for nonnegative tensors and their tropical analogues. Vietnam Journal of Mathematics, 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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