The improvements of Ky Fan theorem are given for tensors. First, based on Brauer-type eigenvalue inclusion sets, we obtain some new Ky Fan-type theorems for tensors. Second, by characterizing the ratio of the smallest and largest values of a Perron vector, we improve the existing results. Third, some new eigenvalue localization sets for tensors are given and proved to be tighter than those presented by Li and Ng (Numer Math 130(2):315–335, 2015) and Wang et al. (Linear Multilinear Algebra 68(9):1817–1834, 2020). Finally, numerical examples are given to validate the efficiency of our new bounds.
<p style='text-indent:20px;'>Generalized minimal Gershgorin set is introduced to locate all generalized tensor eigenvalues. It is proved that this set is tighter than the well-known generalized Gershgorin set. A sequence of approximation sets were also conducted in order to obtain the generalized minimal Gershgorin set and prove that the limit of this sequence is the generalized minimal Gershgorin set. Furthermore, we use numerical examples and figures to illustrate the benefits of our results.</p>
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