In this paper, first we give the definition of standard tensor. Then we clarify the relationship between weakly irreducible tensors and weakly irreducible polynomial maps by the definition of standard tensor. And we prove that the singular values of rectangular tensors are the special cases of the eigen-values of standard tensors related to rectangular tensors. Based on standard tensor, we present a generalized version of the weak Perron-Frobenius Theorem of nonnegative rectangular tensors under weaker conditions. Furthermore, by studying standard tensors, we get some new results of rectangular tensors. Besides, by using the special structure of standard tensors corresponding to nonnegative rectangular tensors, we show that the largest singular value is really geometrically simple under some weaker conditions.
Standard tensor nonnegative rectangular tensor singular value geometrically simple
74B99 15A18 15A69
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The authors would like to thank two referees for their valuable comments and suggestions, which greatly help improve the paper. This work was supported in part by the Natural Science Foundation of Xinjiang Uygur Autonomous Region (Grant No. 2017D01A14).
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