Reducing subspace
In linear algebra, a reducing subspace formula_1 of a linear map formula_2 from a Hilbert space formula_3 to itself is an invariant subspace of formula_4 whose orthogonal complement formula_5 is also an invariant subspace of formula_6 That is, formula_7 and formula_8 One says that the subspace formula_1 reduces the map formula_6 One says that a linear map is reducible if it has a nontrivial reducing subspace. Otherwise one says it is irreducible. If formula_3 is of finite dimension formula_12 and formula_1 is a reducing subspace of the map formula_2 represented under basis formula_15 by matrix formula_16 then formula_17  can be expressed as the sum formula_18 where formula_19 is the matrix of the orthogonal projection from formula_3 to formula_1 and formula_22 is the matrix of the projection onto formula_23 (Here formula_24 is the identity matrix.) Furthermore, formula_3 has an orthonormal basis formula_26 with a subset that is an orthonormal basis of formula_1. If formula_28 is the transition matrix from formula_15 to formula_26 then with respect to formula_26 the matrix formula_32 representing formula_4 is a block-diagonal matrix formula_34 with formula_35 where formula_36, and formula_37