The realization of positive definite matrices via planar networks and mixing-type sub-cluster algebras. (English) Zbl 1463.15073
Summary: 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 [Math. Intell. 22, No. 1, 23–33 (2000; Zbl 1052.15500)], 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.
MSC:
| 15B48 | Positive matrices and their generalizations; cones of matrices |
| 15B57 | Hermitian, skew-Hermitian, and related matrices |
| 13F60 | Cluster algebras |
| 05B20 | Combinatorial aspects of matrices (incidence, Hadamard, etc.) |
| 05C50 | Graphs and linear algebra (matrices, eigenvalues, etc.) |
Keywords:
positive definite matrix; generalized Jacobi matrix; planar network; double wiring diagram; cluster subalgebrasCitations:
Zbl 1052.15500References:
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