Suppose $V$ is a complex vector space with respect to basis $v_{1},...,v_{n}$, and $T: V \rightarrow V$ is a linear transformation with matrix representation $A$.
Now, consider $V$ as a real vector space with respect to basis $v_{1}, iv_{1},...,v_{n}, iv_{n}$. What is matrix representation of $T$ with respect to this basis $v_{j}, iv_{j}$?
My initial idea was trying to represent each "complex $A$" column $Tv_{j} \rightarrow v_{1},...,v_{n}$ as two columns for the "real $A$" matrix such that $Tv_{j} + Tiv_{j} \rightarrow v_{1}, iv_{1},...,v_{n}, iv_{n}$. However, I don't comprehend exactly what the question means by $iv_{j}$ (is it literally $v_{j}$ multiplied by $i$?) or how to represent the new matrix in terms of $A$ (For this, I assume I have to employ something of the form $CAC^{-1}$, where $C$ changes from the real to the complex basis).
I would appreciate some hints on how to proceed from where I am and how to properly interpret the problem.
Thanks!
Let $(v_1,\ldots,v_n)$ be a basis of $V$ in which the matrix of $T$ is $A=(a_{i,j})_{1\leqslant i,j\leqslant n}\in\mathscr{M}_n(\mathbb{C})$ so that $$ T(v_j)=\sum_{k=1}^na_{k,j}v_k=\sum_{k=1}^n{\rm Re}(a_{k,j})v_k+\sum_{k=1}^n{\rm Im}(a_{k,j})iv_k $$ and $$ T(iv_j)=iT(v_j)=-\sum_{k=1}^n{\rm Im}(a_{k,j})v_k+\sum_{k=1}^n{\rm Re}(a_{k,j})iv_k. $$ The matrix of $T$ in the basis $(v_1,iv_1,\ldots,v_n,iv_n)$ is $$ \begin{pmatrix} R_{1,1} & \cdots & R_{1,n} \\ \vdots & & \vdots \\ R_{n,1} & \cdots & R_{n,n} \end{pmatrix}\in\mathscr{M}_{2n,2n}(\mathbb{R}) $$ where $$ R_{i,j}=\begin{pmatrix} {\rm Re}(a_{i,j}) & {\rm -Im}(a_{i,j}) \\ {\rm Im}(a_{i,j}) & {\rm Re}(a_{i,j}) \end{pmatrix}. $$