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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Graphical evaluation of sparse determinants


Authors: Daniel Drucker and David M. Goldschmidt
Journal: Proc. Amer. Math. Soc. 77 (1979), 35-39
MSC: Primary 15A15; Secondary 05C20, 17B20
MathSciNet review: 539626
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Abstract: In this paper, we show that the determinant of a matrix with entries in a commutative ring can be recursively computed by use of an associated directed graph whose circuits are assigned weights in the ring. This result provides an efficient means of calculating the determinants of sparse matrices. As an application, we compute the determinants of the Cartan matrices associated to the simple complex Lie algebras.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1979-0539626-2
PII: S 0002-9939(1979)0539626-2
Keywords: Determinant, sparse matrix, directed graph, circuit, Cartan matrix, Dynkin diagram
Article copyright: © Copyright 1979 American Mathematical Society