# How can I multiply a linear transformation by a antilinear transformation

Well, I have the following inquiry: I have an element $$g \in \mathrm{O}(V)$$. I know that such element can be decomposed as the sum of two operators: $$g=p_g+q_g,$$ where $$p_g$$ is linear and $$q_g$$ is antilinear.

I’m are working over (finite) vector spaces over $$\mathbb{C}$$.

I’m trying to calculate $$T:=q_gp_g^{-1},$$ (assuming that $$p_g$$ is invertible) for a specific $$g$$. I already expressed $$p_g$$ in matrix form and calculated it’s inverse. However, since $$q_g$$ is not linear, it doesn’t hace a matrix representation. ¿How can I calculate $$q_gp_g^{-1}$$? Specifically, $$p\sim \begin{bmatrix} \alpha & -\overline\beta\\ \beta & \overline\alpha \end{bmatrix}$$ and $$q=-\beta \overline{\beta}v+\overline\alpha \beta u$$, where $$\{ v,u\}$$ is a basis of $$V$$ and $$\alpha, \beta \in \mathbb{C}$$. If $$q$$ were a linear transformation, I could put it in matrix form…

What’s the matrix form of an antilinear transformation?

• An antilinear map is still linear over $\mathbb{R}$. Commented Oct 20, 2022 at 2:10
• You mention matrices, so I will assume $\Bbb C^n$. If $C:\Bbb C^n\to\Bbb C^n$ is complex conjugation, then any $\Bbb R$-linear map $A:\Bbb C^n\to\Bbb C^n$ is expressible as $A=A_0+A_1C$ where $A_0$ and $A_1$ are both $\Bbb C$-linear. (Being orthogonal is irrelevant.) Then $A_0(v)=\frac{1}{2i}(iA(v)+A(iv))$ and $A_1(v)=\frac{1}{2i}(iA(v)-A(iv))$. Not sure what form you have $A$ in that you can express $A_0,A_1$ with it. Not sure what conditions on $A_0,A_1$ are equivalent to $A^{-1}$ existing.