Given complex vector spaces $V$, and antilinear $T:V \rightarrow V$, then if we fix a basis of $V$, we can represent $T$ by the matrix of the linear $T \circ J$, where $J$ is complex conjugation.
I would really like to know if we can always find a basis of $V$ where this matrix is in some nice form like the JNF.
If $T$ has matrix $M$, and $A$ is a change of basis matrix, then with respect to the new basis $T$ has matrix $AM\overline{A^{-1}}$, so an equivalent question would be: Given a complex square matrix $M$, can we always find an invertible $A$ such that $AM\overline{A^{-1}}$ is in a nice form.
Many thanks in advance, please let me know if anything is unclear.