In this paper, we focus on some inequalities for the Fan product of $ M $-tensors. Based on Brualdi-type eigenvalue inclusion sets of $ M $-tensors and similarity transformation methods, we establish Brualdi-type inequalities on the minimum eigenvalue for the Fan product of two $ M $-tensors. Furthermore, we discuss the advantages of different Brualdi-type inequalities. Numerical examples verify the validity of the conclusions.

中文翻译：

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M-张量的Fan积的最小特征值的Brualdi型不等式

在本文中，我们关注$ M $张量的Fan乘积的一些不等式。基于$ M $张量的Brualdi型特征值包含集和相似变换方法，我们针对两个$ M $张量的Fan乘积的最小特征值建立了Brualdi型不等式。此外，我们讨论了不同的Brualdi型不等式的优点。数值例子验证了结论的正确性。