Algebraic systems of matrices and Gröbner basis theory

Bourgeois, Gerald

doi:10.1016/j.laa.2008.11.031

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Cited by 2 publications

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“…We may assume that A and N are upper triangular matrices such that N = 0 and, for every i, a ii = λ i . Suppose that (4) for all i = j, λ i − λ j = f ′ (α).…”

Section: General Remarksmentioning

confidence: 99%

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How to solve the matrix equation XA-AX=f(X)

Bourgeois¹

2010

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Let f be an analytic function defined on a complex domain Ω and A ∈ Mn(C). We assume that there exists a unique α satisfying f (α) = 0. When f ′ (α) = 0 and A is non-derogatory, we completely solve the equation XA − AX = f (X). This generalizes Burde's results. When f ′ (α) = 0, we give a method to solve completely the equation XA − AX = f (X): we reduce the problem to solving a sequence of Sylvester equations. Solutions of the equation f (XA − AX) = X are also given in particular cases.

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“…We may assume that A and N are upper triangular matrices such that N = 0 and, for every i, a ii = λ i . Suppose that (4) for all i = j, λ i − λ j = f ′ (α).…”

Section: General Remarksmentioning

confidence: 99%

“…In [6], the author finds exact rational solutions of the equation p(X) = A where p is a polynomial. Finally, the author in [4] found exact solutions of the system {A + B + C = αI n , A 2 + B 2 + C 2 = βI n , A 3 + B 3 + C 3 = γI n }, α, β, γ ∈ C, n ∈ N ≥2 , where the (n × n) matrices A, B, C are to be determined.…”

Section: Introductionmentioning

confidence: 99%