Applications of graph theory to matrix theory
Author:
Frank W. Owens
Journal:
Proc. Amer. Math. Soc. 51 (1975), 242249
MSC:
Primary 15A15
MathSciNet review:
0376708
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Abstract 
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Abstract: Let be matrices over a commutative ring with identity. Graph theoretic methods are established to compute the standard polynomial . It is proved that if , and if the characteristic of either is zero or does not divide , where denotes the greatest integer function, then there exist skewsymmetric matrices such that .
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 S. A. Amitsur and J. Levitzki, Minimal identities for algebras, Proc. Amer. Math. Soc. 1 (1950), 449463. MR 12, 155. MR 0036751 (12:155d)
 [2]
 Joan P. Hutchinson, Doctoral Thesis, University of Pennsylvania, 1973.
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 Bertram Kostant, A theorem of Frobenius, a theorem of AmitsurLevitzki and cohomology theory, J. Math. Mech. 7 (1958), 237264. MR 19, 1153. MR 0092755 (19:1153e)
 [4]
 Frank W. Owens, A graph theoretic generalization of a theorem by Kostant (to appear).
 [5]
 , Matrices with zero diagonal, Notices Amer. Math. Soc. 20 (1973), A7 and A548. Abstract 73TA25.
 [6]
 Louis H. Rowen, Standard polynomials in matrix algebras, Trans. Amer. Math. Soc. 190 (1974), 253284. MR 0349715 (50:2208)
 [7]
 Kirby C. Smith and Hillel J. Kumin, Identities on matrices, Amer. Math. Monthly 79 (1972), 157158. MR 1536623
 [8]
 Richard G. Swan, An application of graph theory to algebra, Proc. Amer. Math. Soc. 14 (1963), 367373. MR 26 #6956. MR 0149468 (26:6956)
 [9]
 , Correction to ``An application of graph theory to algebra", Proc. Amer. Math. Soc. 21 (1969), 379380. MR 41 #101. MR 0255439 (41:101)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939197503767089
PII:
S 00029939(1975)03767089
Keywords:
Standard polynomial,
digraph,
Euler path,
skewsymmetric
Article copyright:
© Copyright 1975
American Mathematical Society
