# American Institute of Mathematical Sciences

doi: 10.3934/jimo.2021146
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## A note on the solvability of a tensor equation

 Department of Mathematics, School of Science, Hangzhou Dianzi University, Hangzhou, 310018, China

Received  January 2021 Revised  May 2021 Early access September 2021

Fund Project: This work is partially supported by National Science Foundation of China (Grant No. 11771328), Young Elite Scientists Sponsorship Program by Tianjin, and the Natural Science Foundation of Zhejiang Province, China (Grant No. LD19A010002)

In this note, we generalize a solvability result for a tensor equation with a nonsingular leading tensor to the case with possibly a singular leading tensor.

Citation: Shenglong Hu. A note on the solvability of a tensor equation. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2021146
##### References:
 [1] S. Friedland, Inverse eigenvalue problem, Linear Algebra Appl., 17 (1977), 15-51.  doi: 10.1016/0024-3795(77)90039-8.  Google Scholar [2] R. C. Gunning and H. Rossi, Analytic Functions of Several Complex Variables, Prentice-Hall, Inc., Englewood Cliffs, NJ, 1965.  Google Scholar [3] S. Hu, Z.-H. Huang, C. Ling and L. Qi, On determinants and eigenvalue theory of tensors, J. Symb. Comput., 50 (2013), 508-531.  doi: 10.1016/j.jsc.2012.10.001.  Google Scholar [4] Z.-H. Huang and L. Qi, Tensor complementarity problems–Part I: Basic theory, J. Optim. Theory Appl., 183 (2019), 1-23.  doi: 10.1007/s10957-019-01566-z.  Google Scholar [5] I. R. Shafarevich, Basic Algebraic Geometry, Translated from the Russian by K. A. Hirsch. Revised printing of Grundlehren der mathematischen Wissenschaften, Vol. 213, 1974. Springer Study Edition. Springer-Verlag, Berlin-New York, 1977.  Google Scholar [6] X. Wang, M. Che and Y. Wei, Existence and uniqueness of positive solution for $H^+$-tensor equations, Appl. Math. Lett., 98 (2019), 191-198.  doi: 10.1016/j.aml.2019.05.046.  Google Scholar [7] X. Wang, M. Che and Y. Wei, Neural network approach for solving nonsingular multi-linear tensor systems, J. Comput. Appl. Math., 368 (2020), 112569. doi: 10.1016/j.cam.2019.112569.  Google Scholar

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##### References:
 [1] S. Friedland, Inverse eigenvalue problem, Linear Algebra Appl., 17 (1977), 15-51.  doi: 10.1016/0024-3795(77)90039-8.  Google Scholar [2] R. C. Gunning and H. Rossi, Analytic Functions of Several Complex Variables, Prentice-Hall, Inc., Englewood Cliffs, NJ, 1965.  Google Scholar [3] S. Hu, Z.-H. Huang, C. Ling and L. Qi, On determinants and eigenvalue theory of tensors, J. Symb. Comput., 50 (2013), 508-531.  doi: 10.1016/j.jsc.2012.10.001.  Google Scholar [4] Z.-H. Huang and L. Qi, Tensor complementarity problems–Part I: Basic theory, J. Optim. Theory Appl., 183 (2019), 1-23.  doi: 10.1007/s10957-019-01566-z.  Google Scholar [5] I. R. Shafarevich, Basic Algebraic Geometry, Translated from the Russian by K. A. Hirsch. Revised printing of Grundlehren der mathematischen Wissenschaften, Vol. 213, 1974. Springer Study Edition. Springer-Verlag, Berlin-New York, 1977.  Google Scholar [6] X. Wang, M. Che and Y. Wei, Existence and uniqueness of positive solution for $H^+$-tensor equations, Appl. Math. Lett., 98 (2019), 191-198.  doi: 10.1016/j.aml.2019.05.046.  Google Scholar [7] X. Wang, M. Che and Y. Wei, Neural network approach for solving nonsingular multi-linear tensor systems, J. Comput. Appl. Math., 368 (2020), 112569. doi: 10.1016/j.cam.2019.112569.  Google Scholar
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