Algebraic systems of matrices and Gröbner basis theory. (English) Zbl 1161.39021
Author’s abstract: The problem of finding all the \(n\times n\) 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 \((\mathbb S)\) in the form: \(\{A+B+C=\alpha I_n,A^{2}+B^{2}+C^{2}=\beta I_n,A^{3}+B^{3}+C^{3}=\gamma I_n\}\) where \(\alpha ,\beta ,\gamma \) are given complex numbers. Except in a special case, we solve explicitly these systems, depending on the values of the parameters \(\alpha ,\beta ,\gamma \). For this purpose, we use Gröbner basis theory. A nilpotent algebra is associated to the special case. The method used for solving \((\mathbb S)\) leads to complete solution of the original problem. We study also similar systems over the \(n\times n\) real matrices and over the skew-field of quaternions.
Reviewer: Sheng Chen (Harbin)
MSC:
| 39B42 | Matrix and operator functional equations |
| 13P10 | Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) |
| 15A30 | Algebraic systems of matrices |
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