Tensor inversion and its application to the tensor equations with Einstein product |
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| Authors: | Mao-lin Liang Rui-juan Zhao |
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| Institution: | 1. School of Mathematics and Statistics, Lanzhou University , Lanzhou, People's Republic of China;2. School of Mathematics and Statistics, Tianshui Normal University , Tianshui, People's Republic of China;3. School of Mathematics and Statistics, Lanzhou University , Lanzhou, People's Republic of China |
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| Abstract: | Recently, the inverse of an even-order square tensor has been put forward in Brazell M, Li N, Navasca C, Tamon C. Solving multilinear systems via tensor inversion. SIAM J Matrix Anal Appl. 2013;34(2):542–570] by means of the tensor group consisting of even-order square tensors equipped with the Einstein product. In this paper, several necessary and sufficient conditions for the invertibility of a tensor are obtained, and some approaches for calculating the inverse (if it exists) are proposed. Furthermore, the Cramer's rule and the elimination method for solving the tensor equations with the Einstein product are derived. In addition, the tensor eigenvalue problem mentioned in Qi L-Q. Theory of tensors (hypermatrices). Hong Kong: Department of Applied Mathematics, The Hong Kong Polytechnic University; 2014] can also be addressed by using the elimination method mentioned above. By the way, the LU decomposition and the Schur decomposition of matrices are extended to tensor case. Numerical examples are provided to illustrate the main results. |
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| Keywords: | Nung-Sing Sze |
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