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title: "invariant subspace"
source: https://www.jemoka.com/posts/kbhinvariant_subspace/
---

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&rsquo;s not \(V\) nor \(\{0\}\) if \(\dim V &gt; 1\) for complex number vector spaces and \(\dim V &gt; 2\) for real number vector spaces.
