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Let $A$ be a nilpotent square matrix, $J$ be the antidiagonal matrix with 1's on the secondary diagonal (i.e. $J^{2}=E$) and let $B=J A J.$ Suppose we conjugate the matrices $A,B$ by a matrix $C:$ $A_1=C A C^{-1}, B_1=C B C^{-1}.$

Question 1. Suppose we know the matrix $A_1$ only (but don't know matrices $A$ and $C.$) Is there any trick (a $C$-invariant construction) to find a matrix $B_1?$

Question 2. Is it possyble (knowing the matrix $A_1$ only) to find the conjugated commutator $C[A,B]C^{-1}$?

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Where does the come from? –  Igor Rivin Sep 23 '11 at 14:36
    
@Igor. It is an original problem and it come from nowhere –  Melania Sep 24 '11 at 6:32
    
You can reformulate the question in terms of $J$, $A_1$, $B_1$ and $C$ only, and the input of $A$ seems irrelevant. Let $J_1=CJC^{-1}, then $B_1=J_1A_1J_1$. So the possibilities for $B_1$ are related to the adjoint orbit of $J$ under the action of an appropriate general linear group. Maybe there is a nice description of this? Similarly, your second question is to find $[A_1,B_1]$, which again is determined up to conjugation by something in the adjoint orbit of $J$. –  David Hill Sep 26 '11 at 20:07
    
@David. Thanks for ansver. Unfortunatelly I dont know any fine description of the class of matrices. –  Melania Sep 28 '11 at 17:13
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