Reducing subspace
In linear algebra, a reducing subspace  of a linear map  from a Hilbert space  to itself is an invariant subspace of  whose orthogonal complement  is also an invariant subspace of  That is,  and  One says that the subspace  reduces the map 

One says that a linear map is reducible if it has a nontrivial reducing subspace. Otherwise one says it is irreducible. If  is of finite dimension  and  is a reducing subspace of the map  represented under basis  by matrix  then  can be expressed as the sum

where  is the matrix of the orthogonal projection from  to  and  is the matrix of the projection onto   (Here  is the identity matrix.) Furthermore,  has an orthonormal basis  with a subset that is an orthonormal basis of . If  is the transition matrix from  to  then with respect to  the matrix  representing  is a block-diagonal matrix

with  where , and

References 

Linear algebra
Matrices