Abstract
The aim of this paper is to give an explicit description of identities satisfied by matrices (n×n over a field k of characteristic 0) in order to be able to compute with formal matrices (“forgetting” their representations with coefficients). We introduce a universal free algebra where all formal manipulations are made. Using classical properties of an ideal of identities in an algebra with trace, we reduce our problem to the study of identities among multilinear traces. These are closely linked with the action of the algebra k[S m] of the symmetric group on the m th tensor product of E=k n. Proving a theorem about the kernel of this action and its effective version, we can decompose all identities of matrices in an explicit way as linear combinations, substitutions, product or traces of the well-known Cayley-Hamilton identity. This leads to an algorithm for reducing to a canonical form modulo the ideal of identities of matrices in the free algebra.
Preview
Unable to display preview. Download preview PDF.
References
S. A. Amitsur, The T-ideals of the free ring, J. Lond. Math. Soc. 30 (1955) 470–475.
F. R. Gantmacher Theory of matrices, (Dunod 1966).
D. E. Knuth, The art of Computer Programming, Vol. 3 (Addison-Wesley 1968).
E. Formanek, Polynomial Identities of matrices, Agebraists'hommage. Papers in ring theory and related topics. Contemp. Math. (1981) 41–79.
J. Pierce, Associative Algebra, (Springer-verlag 1986).
C. Procesi, The invariant theory of n × n matrices, Adv. in Math. 19 (1976) 306–381.
C. Procesi, Computing with 2 × 2 matrices, J. of Alg. 87 (1984) 342–359.
Y. P. Razmyslov, Trace identities of full matrix algebras over a field of characteristic zero, Translation: Math. USSR Izv. 8 (1974) 727–760.
A. Regev, Young Tableaux and P.I. Algebra, Astérisque 87/88 (1981) 335–352.
L. H. Rowen, Polynomials Identities in Ring, (Academic press. New-york 1980).
H. Weyl, The classical groups, (Princeton University Press, Princeton N. J. 1946).
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1991 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Mourrain, B. (1991). Some algebra with formal matrices. In: Sakata, S. (eds) Applied Algebra, Algebraic Algorithms and Error-Correcting Codes. AAECC 1990. Lecture Notes in Computer Science, vol 508. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-54195-0_54
Download citation
DOI: https://doi.org/10.1007/3-540-54195-0_54
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-54195-0
Online ISBN: 978-3-540-47489-0
eBook Packages: Springer Book Archive