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 ) ⊆ W {\displaystyle T(W)\subseteq W} and T ( W ⊥ ) ⊆ W ⊥ . {\displaystyle T(W^{\perp })\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
M = P W M P W + P W ⊥ M P W ⊥ {\displaystyle M=P_{W}MP_{W}+P_{W^{\perp }}MP_{W^{\perp }}}
where P W ∈ R r × r {\displaystyle P_{W}\in \mathbb {R} ^{r\times r}} is the matrix of the orthogonal projection from V {\displaystyle V} to W {\displaystyle W} and P W ⊥ = I − P W {\displaystyle P_{W^{\perp }}=I-P_{W}} is the matrix of the projection onto W ⊥ . {\displaystyle W^{\perp }.} (Here I ∈ R r × r {\displaystyle I\in \mathbb {R} ^{r\times r}} is the identity matrix.) Furthermore, V {\displaystyle V} has an orthonormal basis B ′ {\displaystyle B'} with a subset that is an orthonormal basis of W {\displaystyle W} . If Q ∈ R r × r {\displaystyle Q\in \mathbb {R} ^{r\times r}} is the transition matrix from B {\displaystyle B} to B ′ {\displaystyle B'} then with respect to B ′ {\displaystyle B'} the matrix Q − 1 M Q {\displaystyle Q^{-1}MQ} representing T {\displaystyle T} is a block-diagonal matrix
Q − 1 M Q = [ A 0 0 B ] {\displaystyle Q^{-1}MQ=\left[{\begin{array}{cc}A&0\\0&B\end{array}}\right]}
with A ∈ R d × d , {\displaystyle A\in \mathbb {R} ^{d\times d},} where d = dim W {\displaystyle d=\dim W} , and B ∈ R ( r − d ) × ( r − d ) . {\displaystyle B\in \mathbb {R} ^{(r-d)\times (r-d)}.}
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