Please use this identifier to cite or link to this item: http://hdl.handle.net/10397/25734
Title: On determinants and eigenvalue theory of tensors
Authors: Hu, S
Huang, ZH
Ling, C
Qi, L 
Keywords: Characteristic polynomial
Determinant
Eigenvalue
Tensor
Issue Date: 2013
Source: Journal of symbolic computation, 2013, v. 50, p. 508-531 How to cite?
Journal: Journal of Symbolic Computation 
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 Geršgorin'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.
URI: http://hdl.handle.net/10397/25734
ISSN: 0747-7171
DOI: 10.1016/j.jsc.2012.10.001
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