A note on the replicas of nilpotent matrices
Hsio-Fu Tuan · Bulletin of the American Mathematical Society · 1945
In a recent paper, 1 Chevalley proved the following theorem:(A) If Z is a nilpotent matrix over a field K of characteristic 0, the only replicas Z' of Z are the matrices Z f = tZ, tÇ^K. 2 For the proof of (A), he made use of a particular case of a theorem due to Ado and gave a proof for the results which he needed.In the present note, we shall give a direct simple proof of (A) and we shall in fact deduce it as an immediate consequence of the stronger theorem :(B) If Z and Z' are two nilpotent matrices over afield K of characteristic 0, and if q(x) and r(x) are two polynomials with coefficients in K and without constant terms such that Z' = q(Z) and Zo,2 -r(Zo t 2), then Z' = tZ,t