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

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.

is of finite dimension

is a reducing subspace of the map

represented under basis

can be expressed as the sum

is the matrix of the orthogonal projection from

is the matrix of the projection onto

is the identity matrix.)

has an orthonormal basis

with a subset that is an orthonormal basis of

is the transition matrix from

{\displaystyle Q^{-1}MQ}

is a block-diagonal matrix

{\displaystyle Q^{-1}MQ=\left[{\begin{array}{cc}A&0\\0&B\end{array}}\right]}

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