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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 $W^{\perp}$.

The tangent bundle $TM$ of a manifold $M$ is defined as the disjoint union of all tangent spaces $T_pM$ for every point $p$ in $M$. Symbolically, $TM = \bigsqcup_{p \in M} T_pM$.

Conditions: $M$ is a smooth manifold.; $T_pM$ is the tangent space at point $p$.

The tangent bundle $TM$ of a manifold $M$ is defined as the disjoint union of all tangent spaces $T_pM$ for every point $p$ in $M$. Symbolically, $TM = \bigsqcup_{p \in M} T_pM$.

Conditions: $M$ is a smooth manifold.; $T_pM$ is the tangent space at point $p$.