 Invariant subspace

An invariant subspace of a linear map $A:V\to V$ is a vector subspace $W\subseteq V$ satisfying $A(W)\subseteq W$. For a group representation, invariance means $\rho (g)W\subseteq W$ for every represented group element $g$.

Invariance of a vector subspace does not mean that each individual vector stays fixed. The zero-sum plane in the three-coordinate example is invariant even though the cyclic permutation rotates its nonzero vectors. By contrast, the constant-coordinate line is pointwise fixed. Distinguishing these statements prevents a symmetry-preserved mode space from being mistaken for a static mode.

In functional analysis and operator theory, an **invariant subspace** refers to a subspace of a given vector space that is preserved under the action of a given linear operator. More formally, let \( T: V \to V \) be a linear operator on a vector space \( V \).