Abstract
In this paper, we investigate some inequalities for the Fan product of M-tensors. We propose exact characterizations of M-tensors and establish some inequalities on the minimum eigenvalue for the Fan product of two M-tensors. Furthermore, the inclusion relations among them are discussed. Numerical examples show the validity of the conclusions.
Highlights
Let C(R) be the set of all complex numbers, R+(R++) be the set of all nonnegative numbers, Cn(Rn) be the set of all dimension n complex vectors, and Rn+(Rn++) be the set of all dimension n nonnegative vectors
An mth order ndimensional tensor A = is a higher-order generalization of matrices, which consists of nm entries: ai1i2...im ∈ R, ik ∈ N = {1, 2, . . . , n}, k = 1, 2, . . . , m
Lemma 5 Suppose that P = is a strong M-tensor of order m and dimension n
Summary
Let C(R) be the set of all complex (real) numbers, R+(R++) be the set of all nonnegative (positive) numbers, Cn(Rn) be the set of all dimension n complex (real) vectors, and Rn+(Rn++) be the set of all dimension n nonnegative (positive) vectors. Lemma 1 Let A be a weakly irreducible nonnegative tensor of order m and dimension n. Lemma 2 ([23]) Suppose A is a weakly irreducible Z-tensor and its all diagonal elements are nonnegative.
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