Reducing subspace
In linear algebra, a reducing subspace [math]\displaystyle{ W }[/math] of a linear map [math]\displaystyle{ T:V\to V }[/math] from a Hilbert space [math]\displaystyle{ V }[/math] to itself is an invariant subspace of [math]\displaystyle{ T }[/math] whose orthogonal complement [math]\displaystyle{ W^\perp }[/math] is also an invariant subspace of [math]\displaystyle{ T. }[/math] That is, [math]\displaystyle{ T(W) \subseteq W }[/math] and [math]\displaystyle{ T(W^\perp) \subseteq W^\perp. }[/math] One says that the subspace [math]\displaystyle{ W }[/math] reduces the map [math]\displaystyle{ T. }[/math]
One says that a linear map is reducible if it has a nontrivial reducing subspace. Otherwise one says it is irreducible.
If [math]\displaystyle{ V }[/math] is of finite dimension [math]\displaystyle{ r }[/math] and [math]\displaystyle{ W }[/math] is a reducing subspace of the map [math]\displaystyle{ T:V\to V }[/math] represented under basis [math]\displaystyle{ B }[/math] by matrix [math]\displaystyle{ M \in\R^{r\times r} }[/math] then [math]\displaystyle{ M }[/math] can be expressed as the sum
[math]\displaystyle{ M = P_W M P_W + P_{W^\perp} M P_{W^\perp} }[/math]
where [math]\displaystyle{ P_W \in\R^{r\times r} }[/math] is the matrix of the orthogonal projection from [math]\displaystyle{ V }[/math] to [math]\displaystyle{ W }[/math] and [math]\displaystyle{ P_{W^\perp} = I - P_{W} }[/math] is the matrix of the projection onto [math]\displaystyle{ W^\perp. }[/math][1] (Here [math]\displaystyle{ I \in \R^{r\times r} }[/math] is the identity matrix.)
Furthermore, [math]\displaystyle{ V }[/math] has an orthonormal basis [math]\displaystyle{ B' }[/math] with a subset that is an orthonormal basis of [math]\displaystyle{ W }[/math]. If [math]\displaystyle{ Q \in \R^{r\times r} }[/math] is the transition matrix from [math]\displaystyle{ B }[/math] to [math]\displaystyle{ B' }[/math] then with respect to [math]\displaystyle{ B' }[/math] the matrix [math]\displaystyle{ Q^{-1}MQ }[/math] representing [math]\displaystyle{ T }[/math] is a block-diagonal matrix
[math]\displaystyle{ Q^{-1}MQ = \left[ \begin{array}{cc} A & 0 \\ 0 & B \end{array} \right] }[/math]
with [math]\displaystyle{ A\in\R^{d\times d}, }[/math] where [math]\displaystyle{ d= \dim W }[/math], and [math]\displaystyle{ B\in\R^{(r-d)\times(r-d)}. }[/math]
References
- ↑ R. Dennis Cook (2018). An Introduction to Envelopes : Dimension Reduction for Efficient Estimation in Multivariate Statistics. Wiley. p. 7.
 |  |
