Algebraic systems of matrices and Gröbner basis theory

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The problem of finding all the n × n complex matrices A, B, C such that, for all real t, etA + etB + etC is a scalar matrix reduces to the study of a symmetric system (S) in the form: {A + B + C = α In, A2 + B2 + C2 = β In, A3 + B3 + C3 = γ In} 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. © 2008 Elsevier Inc. All rights reserved.




Bourgeois, G. (2009). Algebraic systems of matrices and Gröbner basis theory. Linear Algebra and Its Applications, 430(8–9), 2157–2169.

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