On determinants and eigenvalue theory of tensors

Authors:
Shenglong Hu;Zheng-Hai Huang;Chen Ling;Liqun Qi
Affiliations:
Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong;Department of Mathematics, School of Science, Tianjin University, Tianjin, China;School of Science, Hangzhou Dianzi University, Hangzhou 310018, China;Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong
Venue:
Journal of Symbolic Computation
Year:
2013

Citing 10
Cited 0

Topics in matrix analysis

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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 Gersgorin@?s 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.