Geodesic
Linear Operators Preserving Similarity Classes and Related Results
Canadian mathematical bulletin, Tome 37 (1994) no. 3, pp. 374-383

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Let Mn be the algebra of n × n matrices over an algebraically closed field of characteristic zero. For A ∊ Mn , denote by the collection of all matrices in Mn that are similar to A. In this paper we characterize those invertible linear operators φ on Mn that satisfy , where for some given A1,..., Ak ∊ Mn and denotes the (Zariski) closure of S. Our theorem covers a result of Howard on linear operators mapping the set of matrices annihilated by a given polynomial into itself, and extends a result of Chan and Lim on linear operators commuting with the function f(x) = xk for a given positive integer k ≥ 2. The possibility of weakening the invertibility assumption in our theorem is considered, a partial answer to a conjecture of Howard is given, and some extensions of our result to arbitrary fields are discussed.
DOI : 10.4153/CMB-1994-055-0
Mots-clés : 15A42 
Li, Chi-Kwong; Pierce, Stephen. Linear Operators Preserving Similarity Classes and Related Results. Canadian mathematical bulletin, Tome 37 (1994) no. 3, pp. 374-383. doi: 10.4153/CMB-1994-055-0
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     title = {Linear {Operators} {Preserving} {Similarity} {Classes} and {Related} {Results}},
     journal = {Canadian mathematical bulletin},
     pages = {374--383},
     year = {1994},
     volume = {37},
     number = {3},
     doi = {10.4153/CMB-1994-055-0},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-1994-055-0/}
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