# Analogy of Jordon Normal Form for Antilinear Maps

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.

• Right that was me being stupid, I'll edit the question! – James Jul 20 at 22:17
• A few further comments - when you say "fix a basis for $V$", do you mean a real basis or a complex one? If you mean a complex basis, then you can't really represent $T$ with respect to such a basis as $T$ is not complex linear -- maybe you can do that but you need to specify what you want. If you mean a real basis, then you can always forget the fact that $T$ is antilinear and use the real jordan canonical form. If $V = \mathbb{C}^n$ then you have a "canonical conjugation" $\sigma$ – levap Jul 20 at 22:23
• (don't use $J$ for compex conjugation, $J$ is used for the complex structure -- multiplication by $i$) and you can consider $T \circ \sigma$ which is $\mathbb{C}$-linear and represent $T \circ \sigma$ with respect to a complex basis in Jordan form. – levap Jul 20 at 22:23