Details:
Title  On determinants and eigenvalue theory of tensors  Author(s)  Shenglong Hu, ZhengHai Huang, Chen Ling, Liqun Qi  Type  Article in Journal  Abstract  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 kth order trace of a tensor is equal to the sum of the kth 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.  Keywords  Tensor, Eigenvalue, Determinant, Characteristic polynomial  ISSN  07477171 
URL 
http://www.sciencedirect.com/science/article/pii/S074771711200168X 
Language  English  Journal  Journal of Symbolic Computation  Volume  50  Number  0  Pages  508  531  Year  2013  Edition  0  Translation 
No  Refereed 
No 
