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
[1] (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
![{\displaystyle Q^{-1}MQ=\left[{\begin{array}{cc}A&0\\0&B\end{array}}\right]}](//wikimedia.org/api/rest_v1/media/math/render/svg/b61f34dffb33ef67b911689ef296b69cb30e9160)
with
where
, and 