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 [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

with where , and

References

  1. ^ R. Dennis Cook (2018). An Introduction to Envelopes : Dimension Reduction for Efficient Estimation in Multivariate Statistics. Wiley. p. 7.