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The realization of positive definite matrices via planar networks and mixing-type sub-cluster algebras

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Abstract

As an improvement of the combinatorial realization of totally positive matrices via the essential positive weightings of certain planar network by S. Fomin and A. Zelevinsky [7], in this paper, we give a test method of positive definite matrices via the planar networks and the so-called mixing-type sub-cluster algebras respectively, introduced here originally. This work firstly gives a combinatorial realization of all matrices through planar network, and then sets up a test method for positive definite matrices by LDU-decompositions and the horizontal weightings of all lines in their planar networks. On the other hand, mainly the relationship is built between positive definite matrices and mixing-type sub-cluster algebras.

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Acknowledgements

The first author is supported by the National Natural Science Foundation of China (No. 11671350 and No. 11571173). This work was carried out when she was a Ph.D. candidate at Zhejiang University. The second author is supported by the National Natural Science Foundation of China (No. 11801043) and Natural Science Foundation for Youths of Jiangsu Province (No. BK20181031). Both authors would like to thank Prof. Fang Li for his helpful discussions and warm hospitality.

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Authors and Affiliations

  1. Department of Mathematics, Zhejiang University, Yuquan Campus, Hangzhou, 310027, China

    Diana Ahmad

  2. Department of Mathematics, Changshu Institute of Technology, Changshu, 215500, China

    Yi-chao Yang

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  1. Diana Ahmad
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  2. Yi-chao Yang
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Correspondence to Diana Ahmad or Yi-chao Yang.

Additional information

Supported by the National Natural Science Foundation of China (11671350, 11571173, 11801043) and Natural Science Foundation for Youths of Jiangsu Province (BK20181031).

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Ahmad, D., Yang, Yc. The realization of positive definite matrices via planar networks and mixing-type sub-cluster algebras. Appl. Math. J. Chin. Univ. 35, 127–140 (2020). https://doi.org/10.1007/s11766-020-3617-1

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  • DOI: https://doi.org/10.1007/s11766-020-3617-1

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