The Matrix Equation XA- AX=Xαg(X) over Fields or Rings. (1st October 2014)

• Record Type: Journal Article
• Authors: Bourgeois, Gerald
• Other Names: Li Zhongshan Academic Editor.
• Is Part Of: Algebra. Volume 2014(2014)
• Publication Date: 2014-10-01
• Subjects: Algebra -- Periodicals; Algebra; Electronic journals; Periodicals; 512.005
• DOI: 10.1155/2014/745029
• Languages: English
• ISSNs: 2314-4106

Abstract

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 .