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The block-diagonal matrix form represents the decomposition of the representation $\rho$ into independent subrepresentations on the invariant subspaces $W$ and $W^{\perp}$. When $V = W \oplus W^{\perp}$ and both summands are $G$-invariant, the matrix of $\rho(g)$ in a basis adapted to this direct sum has zero off-diagonal blocks, meaning the action on $W$ does not mix with the action on $W^{\perp}$.

Conditions: $V = W \oplus W^{\perp}$ is a direct sum decomposition.; $W$ and $W^{\perp}$ are both $G$-invariant subspaces.; A basis for $V$ is chosen as the union of bases for $W$ and $W^{\perp}$.

The block-diagonal matrix form represents the decomposition of the representation $\rho$ into independent subrepresentations on the invariant subspaces $W$ and $W^{\perp}$. When $V = W \oplus W^{\perp}$ and both summands are $G$-invariant, the matrix of $\rho(g)$ in a basis adapted to this direct sum has zero off-diagonal blocks, meaning the action on $W$ does not mix with the action on $W^{\perp}$.

Conditions: $V = W \oplus W^{\perp}$ is a direct sum decomposition.; $W$ and $W^{\perp}$ are both $G$-invariant subspaces.; A basis for $V$ is chosen as the union of bases for $W$ and $W^{\perp}$.