Abstract
In this paper, we give a new Z-eigenvalue localization set for Z-eigenvalues of structured fourth order tensors. As applications, a sharper upper bound for the Z-spectral radius of weakly symmetric nonnegative fourth order tensors is obtained and a new Z-eigenvalue based sufficient condition for the positive definiteness of fourth order tensors is also presented. Finally, numerical examples are given to verify the efficiency of our results.
Highlights
Let A = ∈ R[m,n] be an m-th order n dimensional real square tensor, x be a real n-vector.let N = {1, 2, . . . , n}, we define the following real n-vector: Ax m −1 ! n = ∑ i2,···,im =1, x [m−1] =i∈ N .aii2 ...im xi2 . . . xim i∈ NIf there exists a real vector x and a real number λ such thatA x m−1 = λx [m−1], λ is called H-eigenvalue of A and x is called H-eigenvector of A associated with λ
We obtain a sharp upper bound for weakly symmetric nonnegative tensors
We provide a new checkable sufficient condition for the positive definiteness of fourth order tensors, which is based on the inclusion set for Z-eigenvalues of structured fourth order tensors
Summary
Provided some sufficient conditions for the positive definiteness of an even-order real symmetric tensor [3], and some improved results are obtained in [4,5,6,7,8]. Let A be an even-order real symmetric tensor with all positive diagonal entries. Much literature has focused on the properties of Z-eigenvalues of tensors [15,16,17,18,19,20,21,22,23,24], but there are no Z-eigenvalues based sufficient conditions for the positive definiteness of an even-order real symmetric tensor. In this paper, based on the Z-eigenvalue localization sets of structured fourth order tensors, a new sufficient condition for the positive definiteness of fourth order tensors is given
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