= Solution
A <vector subspace> $W\subseteq\mathbb C^n$ is an <invariant subspace> for the <group representation> $D$ if
$$
\boxed{D(g)W\subseteq W\quad\text{for every }g\in G}.
$$
Because $g^{-1}$ is also in the <group> and $D(g)$ is invertible, the inclusion is equivalently $D(g)W=W$ for every $g$.
For $n>0$, the representation is an <irreducible representation> when its only invariant <vector subspaces> are
$$
\boxed{W=\{0\}\quad\text{and}\quad W=\mathbb C^n}.
$$
Thus no nonzero proper subspace is preserved by all representing <matrices>.
Back to article page
