On some properties of the determinants of tensors

Hankel tensors, Vandermonde tensors and their positivities

“…The singularity of a square matrix can also be checked by its spectrum, that is, A is singular if and only if A has zero as one of its eigenvalues. This also applies to the high order tensors due to Hu [7] and the recent work by Shao [14] where it is shown that the determinant of a cubic tensor can be written as a product of some powers of eigenvalues of A. There are many definitions of singularities of tensors.…”

Brualdi-type eigenvalue inclusion sets of tensors

“…Let A and B be order m 2 and order k 1, dimension n tensors, respectively. The product AB is the following tensor C of order (m −1)(k −1) +1 and dimension n with entries: [15].) Let A be an order m dimension n tensor, and let B be an order [13].)…”

Tensors product and hyperdeterminant of boundary formats product

“…Since Qi and Lim introduced the definitions of various types of eigenvalues and eigenvectors of tensors, many researchers pay extensive attention to this area, and quite many interesting research works in this area appeared recently (see [2,4,3,7,6,8] and so on).…”

“…Recently, motivated by the study of eigenvalues of tensors, many results about determinant of tensors appeared (see [7,10,11] and their references). Specially, in 2013, Shao [8] showed the following definition of tensors product: Definition 1.2.…”

“…Applying Definition 1.2, Shao and others [7] got the formula for hyperdeterminant of tensors product as following:…”