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