Maschke's theorem states that if $G$ is a finite group and $V$ a nonzero $G$-module over $\mathbb{C}$ then $$ V=W_1\oplus W_2\oplus ... \oplus W_k $$ where $W_i$ are irreducible submodules of $V$. There is a similar matrix variant of Maschke's theorem, and I want to know if there is anything wrong with my following argumentation.
Let $V$ be the $G$-module such that $$ g\textbf{v}=X(g)\textbf{v} $$ If we choose a matrix of transformation $T$ to be the one that expresses each $X(g)$ in the basis $$ B_V=\{ B_{W_1}, B_{W_2}, ..., B_{W_k} \} $$ with $B_{W_i}$ being a basis for irreducible submodule $W_i$, then we can write $$ TX(g)T^{-1}=\left(\begin{array}{cccc}X_1(g)&0&\cdots&0\\ 0&X_2(g)&\cdots&0\\ \vdots&\vdots&\ddots&\vdots\\ 0&0&\cdots&X_k(g)\end{array}\right), $$ where $X_i(g)$ is the matrix representation of $g$ afforded by submodule $W_i$ in the basis $B_{W_i}$.