V-singular values of rectangular tensors and their applications

Rectangular M-tensors and strong rectangular M-tensors

“…is the largest V-singular value of . The following Perron-Frobenius theorem for V-singular values of nonnegative square rectangular tensors is introduced in [2].…”

“…is a partially symmetric rectangular tensor, p and q are even. Then, there exist V-singular value of and associated left and right eigenvectors [2].…”

“…Definition 2 [2] Let ∈ [p;q;m;n] be a partially symmetric rectangular tensor, p, q 2. If there exist a number λ ∈ , vectors x ∈ m \{0}, and…”

“…is strictly copositive if and only if N2 min ( ) ≡ min{ x p y q : x ∈ m + , y ∈ n + , > 0. (10)Proof : For any 0 = x ∈ m + , 0 = y ∈ n + 1 min ( ) 0 if and only if N 2 min ( ) 0.…”

Structured Rectangular Tensors and Rectangular Tensor Complementarity Problems

“…Let A ∈ R [p;q;m;n] be a partially symmetric rectangular tensor, and p and q are even. en, A is positive definite if and only if all of its H-singular values (or V-singular values) are positive [13][14][15][16][17][18][19]. e definition of copositive rectangular tensors is introduced in [20], which can be viewed as a generalization of copositive square tensors, and some necessary and sufficient conditions for a real partially symmetric rectangular tensor to be a copositive rectangular tensor are also given in [20].…”