Conjugate Matrix

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 2011 at 14:36 @Igor. It is an original problem and it come from nowhere – Melania Sep 24 2011 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 2011 at 20:07 @David. Thanks for ansver. Unfortunatelly I dont know any fine description of the class of matrices. – Melania Sep 28 2011 at 17:13