Show that $U$ and $W$ are subspaces of $V$ which are invariant under $\varepsilon$, that is $U \varepsilon \subset U, W \varepsilon \subset W$.
Prove that
$$
V=U \oplus W
$$
Deduce that if $E$ is an $m \times m$ matrix such that
$$
E^{2}=E, \quad E \neq 0, \quad E \neq I,
$$
there exists an integer $r$ satisf ying $1 \leqslant r<m$ and a non-singular matrix $T$ such that
$$
T^{-1} E T=\left(\begin{array}{ll}
I_{r} & 0 \
0 & 0
\end{array}\right)=J
$$
6CCM351A COURSE NOTES ：
Next we construct the matrix
$$
C=C(\Xi)=\sum_{y \in G} B\left(y^{-1}\right) \Xi A(y) .
$$
Let $x$ be a fixed element of $G$; then $z=y x$ ranges over $G$ when $y$ does. Thus we may equally well write
$$
C=\sum_{z} B\left(z^{-1}\right) \Xi A(z)=\sum_{y} B\left(x^{-1} y^{-1}\right) \Xi A(y x) .
$$
Since $A$ and $B$ are representations, we obtain that
$$
C=B\left(x^{-1}\right)\left(\sum_{y} B\left(y^{-1}\right) \Xi A(y)\right) A(x),
$$
or
$$
B(x) C=C A(x) \quad(x \in G),
$$