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TitleAlgebraic systems of matrices and Gröbner basis theory
Author(s) Gerald Bourgeois
TypeArticle in Journal
AbstractThe problem of finding all the n × n complex matrices A , B , C such that, for all real t , e tA + e tB + e tC is a scalar matrix reduces to the study of a symmetric system ( S ) in the form: A + B + C = α I n , A 2 + B 2 + C 2 = β I n , A 3 + B 3 + C 3 = γ I n where α , β , γ are given complex numbers. Except in a special case, we solve explicitly these systems, depending on the values of the parameters α , β , γ . For this purpose, we use Gröbner basis theory. A nilpotent algebra is associated to the special case. The method used for solving ( S ) leads to complete solution of the original problem. We study also similar systems over the n × n real matrices and over the skew-field of quaternions.
KeywordsSystems of matrices, Gröbner basis, Quaternions, Nilpotent semigroup
URL http://www.sciencedirect.com/science/article/pii/S0024379508005570
JournalLinear Algebra and its Applications
Pages2157 - 2169
Translation No
Refereed No