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 \( V \).
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An invariant subspace of a linear map is a vector subspace satisfying . For a group representation, invariance means for every represented group element .
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.