Reducing subspace

{{Short description|Concept in linear algebra}}

In linear algebra, a reducing subspace $W$ of a linear map $T:V\backslash to\; V$ from a Hilbert space $V$ to itself is an invariant subspace of $T$ whose orthogonal complement $W^\backslash perp$ is also an invariant subspace of $T.$ That is, $T(W)\; \backslash subseteq\; W$ and $T(W^\backslash perp)\; \backslash subseteq\; W^\backslash perp.$ 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 $T:V\backslash to\; V$ represented under basis $B$ by matrix $M\; \backslash in\backslash R^\{r\backslash times\; r\}$ then $M$ can be expressed as the sum

$$M\; =\; P\_W\; M\; P\_W\; +\; P\_\{W^\backslash perp\}\; M\; P\_\{W^\backslash perp\}$$

where $P\_W\; \backslash in\backslash R^\{r\backslash times\; r\}$ is the matrix of the orthogonal projection from $V$ to $W$ and $P\_\{W^\backslash perp\}\; =\; I\; -\; P\_\{W\}$ is the matrix of the projection onto $W^\backslash perp.${{cite book|author=R. Dennis Cook|title=An Introduction to Envelopes : Dimension Reduction for Efficient Estimation in Multivariate Statistics|publisher=Wiley|year=2018|page=7}} (Here $I\; \backslash in\; \backslash R^\{r\backslash times\; r\}$ is the identity matrix.)

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

with $A\backslash in\backslash R^\{d\backslash times\; d\},$ where $d=\; \backslash dim\; W$, and $B\backslash in\backslash R^\{(r-d)\backslash times(r-d)\}.$