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.
For a linear group representation, the common fixed vectors of a subgroup form an invariant subspace for any equivariant map restricted to it. Its dimension is central to symmetry-based branching theorems. It need not be invariant under the entire group; conjugate subgroups have conjugate fixed-point subspaces.
A nonzero representation is reducible when it has a nonzero proper invariant subspace; otherwise it is an irreducible representation.
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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 \( V \).