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@jemoka / Jemoka Knowledge Base / wiki/concepts/invariant_subspace.md
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--- title: "invariant subspace" type: concept related: [Real Number, Invariant Subspace, Subspace, Complex Number, Vector Space] source: https://www.jemoka.com/posts/kbhinvariant_subspace/ confidence: high status: active --- 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.