Theory and Computation of Complex Tensors and its Applications by Maolin Che & Yimin Wei

Author:Maolin Che & Yimin Wei
Language: eng
Format: epub
ISBN: 9789811520594
Publisher: Springer Singapore

for all i n, j n and n. When is diagonalizable and symmetric, Che et al. [44] investigate the inequalities on .

5.2 Sign Nonsingular Tensors

If there exists with n = 2, 3, …, N such that

where K is given in (5.1.2), then α is called a nonzero term in (5.1.1). The sign pattern of , denoted by , belongs to the set RT N,I with entries

Let be the set of tensors with the same sign pattern of . If and have the same sign for any , then we say that has a signed determinant. If for any , then is called a sign nonsingular tensor (differently defined in [45]). Note that singularity and sign-singularity of tensors do not imply each other. The following lemma extends [46, Lemma 1.2.4] from matrices to tensors.

Lemma 5.2.1

Suppose that with an even N. Then the tensor has a signed determinant if and only if all nonzero terms in (5.1.1) have the same sign.

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