کد مقاله | کد نشریه | سال انتشار | مقاله انگلیسی | نسخه تمام متن |
---|---|---|---|---|
401590 | 675389 | 2013 | 24 صفحه PDF | دانلود رایگان |

We investigate properties of the determinants of tensors, and their applications in the eigenvalue theory of tensors. We show that the determinant inherits many properties of the determinant of a matrix. These properties include: solvability of polynomial systems, product formula for the determinant of a block tensor, product formula of the eigenvalues and Geršgorinʼs inequality. As a simple application, we show that if the leading coefficient tensor of a polynomial system is a triangular tensor with nonzero diagonal elements, then the system definitely has a solution in the complex space. We investigate the characteristic polynomial of a tensor through the determinant and the higher order traces. We show that the k-th order trace of a tensor is equal to the sum of the k-th powers of the eigenvalues of this tensor, and the coefficients of its characteristic polynomial are recursively generated by the higher order traces. Explicit formula for the second order trace of a tensor is given.
Journal: Journal of Symbolic Computation - Volume 50, March 2013, Pages 508-531