Characteristic polynomial and higher order traces of third order three dimensional tensors |
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Authors: | Guimei ZHANG Shenglong HU |
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Institution: | 1. School of Mathematics, Tianjin University, Tianjin 300350, China2. Department of Mathematics, School of Science, Hangzhou Dianzi University, Hangzhou 310018, China |
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Abstract: | Eigenvalues of tensors play an increasingly important role in many aspects of applied mathematics. The characteristic polynomial provides one of a very few ways that shed lights on intrinsic understanding of the eigenvalues. It is known that the characteristic polynomial of a third order three dimensional tensor has a stunning expression with more than 20000 terms, thus prohibits an effective analysis. In this article, we are trying to make a concise representation of this characteristic polynomial in terms of certain basic determinants. With this, we can successfully write out explicitly the characteristic polynomial of a third order three dimensional tensor in a reasonable length. An immediate benefit is that we can compute out the third and fourth order traces of a third order three dimensional tensor symbolically, which is impossible in the literature. |
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Keywords: | Tensor traces characteristic polynomial |
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