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In linear algebra, a reducing subspace W {\displaystyle W} of a linear map T : V → V {\displaystyle T:V\to V} from a Hilbert space V {\displaystyle V} to itself is an invariant subspace of T {\displaystyle T} whose orthogonal complement W ⊥ {\displaystyle W^{\perp }} is also an invariant subspace of T . {\displaystyle T.} That is, T ⊆ W {\displaystyle T\subseteq W} and T ⊆ W ⊥ . {\displaystyle T\subseteq W^{\perp }.} One says that the subspace W {\displaystyle W} reduces the map T . {\displaystyle T.}
One says that a linear map is reducible if it has a nontrivial reducing subspace. Otherwise one says it is irreducible.
If V {\displaystyle V} is of finite dimension r {\displaystyle r} and W {\displaystyle W} is a reducing subspace of the map T : V → V {\displaystyle T:V\to V} represented under basis B {\displaystyle B} by matrix M ∈ R r × r {\displaystyle M\in \mathbb {R} ^{r\times r}} then M {\displaystyle M} can be expressed as the sum
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