Invariant subspace

In mathematics, an invariant subspace of a linear mapping T : V → V from some vector space V to itself is a subspace W of V that is preserved by T; that is, T(W) ⊆ W.

Contents

1 General description

1.1 Trivial examples of invariant subspaces
1.2 Uni-dimensional invariant subspace U

2 Formal description
3 Matrix representation
4 Invariant subspace problem
5 Invariant-subspace lattice
6 Fundamental theorem of noncommutative algebra
7 Left ideals
8 See also
9 Bibliography

General description[edit]
Consider a linear mapping

T

{\displaystyle T}

that transforms:

T
:

R

n

R

n

{\displaystyle T:\mathbb {R} ^{n}\to \mathbb {R} ^{n}}

An invariant subspace

W

{\displaystyle W}

of

T

{\displaystyle T}

has the property that all vectors

v

W

{\displaystyle v\in W}

are transformed by

T

{\displaystyle T}

into vectors also contained in

W

{\displaystyle W}

. This can be stated as

v

W

T
(
v
)

W

{\displaystyle v\in W\Rightarrow T(v)\in W}

Trivial examples of invariant subspaces[edit]

R

n

{\displaystyle \mathbb {R} ^{n}}

: Since

T

{\displaystyle T}

maps every vector in

R

n

{\displaystyle \mathbb {R} ^{n}}

into

R

n

{\displaystyle \mathbb {R} ^{n}}

{
0
}

{\displaystyle \{0\}}

: Since a linear map has to map

0

0

{\displaystyle 0\to 0}

Uni-dimensional invariant subspace U[ed