Some variational principles for Z-eigenvalues of nonnegative tensors

Many important spectral properties of nonnegative matrices have recently been successfully extended to higher order nonnegative tensors; for example, see (Chang et al., 2008, 2011; Friedland et al., in press; Lim, 2005; Liu et al., 2010; Ng et al., 2010; Qi et al., 2007; Yang and Yang, 2010) [2,3,9,17,23,24,27,28]. However, most of these results focus on the H-eigenvalues introduced by Qi (2005, 2007) [25,26]. The key results of this paper reveal some similarities as well as some crucial differences between Z-eigenvalues and H-eigenvalues of a nonnegative tensor. In particular, neither the positive Z-eigenvalue nor the associated positive Z-eigenvector of an irreducible nonnegative tensor has to be unique in general as demonstrated by Example 2.7. Furthermore, the Collatz type min–max characterizations of the largest positive Z-eigenvalue of an irreducible nonnegative tensor is only partially true in general as seen in Theorem 4.7 and Example 4.8.