Please use this identifier to cite or link to this item: http://hdl.handle.net/10397/16274
Title: Eigenvalues and invariants of tensors
Authors: Qi, L 
Keywords: Eigenvalue
Invariant
Rank
Supermatrix
Tensor
Issue Date: 2007
Publisher: Academic Press
Source: Journal of mathematical analysis and applications, 2007, v. 325, no. 2, p. 1363-1377 How to cite?
Journal: Journal of mathematical analysis and applications 
Abstract: A tensor is represented by a supermatrix under a co-ordinate system. In this paper, we define E-eigenvalues and E-eigenvectors for tensors and supermatrices. By the resultant theory, we define the E-characteristic polynomial of a tensor. An E-eigenvalue of a tensor is a root of the E-characteristic polynomial. In the regular case, a complex number is an E-eigenvalue if and only if it is a root of the E-characteristic polynomial. We convert the E-characteristic polynomial of a tensor to a monic polynomial and show that the coefficients of that monic polynomial are invariants of that tensor, i.e., they are invariant under co-ordinate system changes. We call them principal invariants of that tensor. The maximum number of principal invariants of mth order n-dimensional tensors is a function of m and n. We denote it by d (m, n) and show that d (1, n) = 1, d (2, n) = n, d (m, 2) = m for m ≥ 3 and d (m, n) ≤ mn - 1 + ⋯ + m for m, n ≥ 3. We also define the rank of a tensor. All real eigenvectors associated with nonzero E-eigenvalues are in a subspace with dimension equal to its rank.
URI: http://hdl.handle.net/10397/16274
ISSN: 0022-247X
EISSN: 1096-0813
DOI: 10.1016/j.jmaa.2006.02.071
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