Abstract
The positive definiteness of rectangular tensors has wide applications in solid mechanics and quantum physics. By modifying the existing definition of singular value for rectangular tensors, some V-singular value inclusion sets for rectangular tensors with positive diagonal entries are established to provide verifiable sufficient conditions for the positive definiteness of rectangular tensors. In addition, an upper bound for the largest V-singular value of nonnegative rectangular tensors is provided.
Highlights
We give an upper bound for the largest V-singular value of nonnegative rectangular tensors, and sufficient conditions of the positive definiteness for rectangular tensors are provided in Sect
Let R(C) be the real field, p, q, m, n be positive integers, m, n ≥ 2, [n] = {1, 2, . . . , n}
2 V-singular values of rectangular tensors First, we introduce the definition of a V-singular value for a rectangular tensor, which can be viewed as a variant of the definition of H-singular value [1, 2]
Summary
We give an upper bound for the largest V-singular value of nonnegative rectangular tensors, and sufficient conditions of the positive definiteness for rectangular tensors are provided in Sect. 3 V-singular value inclusion sets for rectangular tensors with positive diagonal entries Based on Theorem 5, an upper bound for the largest V-singular value of nonnegative rectangular tensors is given.
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