SUMMARYTwo new eigenvalue inclusion sets for tensors are established. It is proved that the new eigenvalue inclusion sets are tighter than that in Qi's paper “Eigenvalues of a real supersymmetric tensor”. As applications, upper bounds for the spectral radius of a nonnegative tensor are obtained, and it is proved that these upper bounds are sharper than that in Yang's paper “Further results for Perron–Frobenius theorem for nonnegative tensors”. And some sufficient conditions of the positive definiteness for an even‐order real supersymmetric tensor are given. Copyright © 2012 John Wiley & Sons, Ltd.

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An S-type eigenvalue localization set for a tensor is given by breaking N = {1, 2, · · · , n} into disjoint subsets S and its complement. It is shown that the new set is tighter than those provided by L.

A new eigenvalue localization set for tensors is given, and proved to be tighter than those in [Qi L. Eigenvalues of a real supersymmetric tensor. J. Symbolic Comput. 2005;40:1302-1324 and [Li CQ, Li YT, Kong X. New eigenvalue inclusion sets for tensors. Numer. Linear Algebra Appl. 2014;21:39-50]. Based on this set, we give two checkable sufficient conditions of the positive definiteness of tensors, and two checkable sufficient conditions of the positive semi-definiteness of tensors.

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A new tensor eigenvalue inclusion set is given, and proved to be tighter than those in L.Q. Qi (2005) [18] and C.Q. Li, Y.T. Li, X. Kong (2014) [12]. In addition, we study the eigenvalues lying on the boundary of the eigenvalue inclusion set provided by Qi (2005) for weakly irreducible tensors. As applications, we give new bounds for the spectral radius of nonnegative tensors, some sufficient conditions for a tensor to be a strong M -tensor and some inequalities to identify the positive definiteness for an even-order real supersymmetric tensor.

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