Reducing subspace explained

In linear algebra, a reducing subspace

W of a linear map T:V\to V from a Hilbert space V to itself is an invariant subspace of T whose orthogonal complement W ^{\perp} is also an invariant subspace of T. That is, T(W) \subseteq W and T(W ^{\perp)} \subseteq W ^{\perp.} One says that the subspace W reduces the map T.

One says that a linear map is reducible if it has a nontrivial reducing subspace. Otherwise one says it is irreducible.

If

V is of finite dimension r and W is a reducing subspace of the map T:V\to V represented under basis B by matrix M \in\R ^{r x} then M can be expressed as the sum

$$M\; =\; P\_W\; M\; P\_W\; +\; P\_\; M\; P\_$$

where

P _{W} \in\R ^{r x} is the matrix of the orthogonal projection from V to W and P _{W ^{\perp}} = I - P _{W} is the matrix of the projection onto W ^{\perp.} ^[1] (Here I \in \R ^{r x} is the identity matrix.)

Furthermore,

V has an orthonormal basis B' with a subset that is an orthonormal basis of W . If Q \in \R ^{r x} is the transition matrix from B to B' then with respect to B' the matrix Q ^{-1} MQ representing T is a block-diagonal matrix

with

A\in\R ^{d x} , where d = \dim W , and B\in\R ^{(r-d) x (r-d)} .

Notes and References

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