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
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.