Remote Access Proceedings of the American Mathematical Society
Green Open Access

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
DOI: https://doi.org/10.1090/S0002-9939-1979-0539626-2
MathSciNet review: 539626
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)

  • [1] N. Bourbaki, Éléments de mathematique, Fasc. 34, Groupes et algèbres de Lie, Chapitres IV, V, VI, Actualités Sci. Indust., no. 1337 Hermann, Paris, 1968. MR 0240238 (39:1590)
  • [2] K. Hoffman and R. Kunze, Linear algebra, 2nd ed., Prentice-Hall, Englewood Cliffs, N. J., 1971. MR 0276251 (43:1998)
  • [3] J. E. Humphreys, Introduction to Lie algebras and representation theory, Springer-Verlag, Berlin and New York, 1972. MR 0323842 (48:2197)
  • [4] N. Jacobson, Lie algebras, Interscience, New York, 1962. MR 0143793 (26:1345)
  • [5] J.-P. Serre, Algèbres de Lie semi-simples complexes, Benjamin, New York, 1966. MR 0215886 (35:6721)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 15A15, 05C20, 17B20

Retrieve articles in all journals with MSC: 15A15, 05C20, 17B20


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1979-0539626-2
Keywords: Determinant, sparse matrix, directed graph, circuit, Cartan matrix, Dynkin diagram
Article copyright: © Copyright 1979 American Mathematical Society

American Mathematical Society