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The Matrix Equation X A - A X = X α g ( X ) over Fields or Rings

Abstract : Let n , α ∈ N ≥ 2 and let K be an algebraically closed field with characteristic 0 or greater than n . We show that if f ∈ K [ X ] and A , B ∈ M n ( K ) satisfy [ A , B ] = f ( A ) , then A , B are simultaneously triangularizable. Let R be a reduced ring such that n ! is not a zero divisor and let A be a generic matrix over R ; we show that X = 0 is the sole solution of A X - X A = X α . Let R be a commutative ring with unity; let A be similar to d i a g ( λ 1 I n 1 , … , λ r I n r ) such that, for every i ≠ j , λ i - λ j is not a zero divisor. If X is a nilpotent solution of X A - A X = X α g ( X ) where g ∈ R [ X ] , then A X = X A .
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Submitted on : Friday, November 5, 2021 - 11:01:10 PM
Last modification on : Monday, November 15, 2021 - 7:30:02 PM

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Gerald Bourgeois. The Matrix Equation X A - A X = X α g ( X ) over Fields or Rings. Algebra Colloquium, World Scientific Publishing, 2014, 2014, pp.1-6. ⟨10.1155/2014/745029⟩. ⟨hal-03417890⟩

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