invariant subspaces are a property of operators; it is a subspace for which the operator in question on the overall space is also an operator of the subspace. constituents an operator T \in \mathcal{L}(V) a subspace U \subset V requirements U is considered invariant on T if u \in U \implies Tu \in U (i.e. U is invariant under T if T |_{U} is an operator on U) additional information nontrivial invariant subspace (i.e. eigenstuff) A proof is not given yet, but T \in \mathcal{L}(V) has an invariant subspace that’s not V nor \{0\} if \dim V > 1 for complex number vector spaces and \dim V > 2 for real number vector spaces.