Invariant subspace

Invariant subspace $W$ ${\ displaystyle W}$ $W$ vector space $V$ ${\ displaystyle V}$ $V$ regarding linear display $T\colon V\to V$ ${\ displaystyle T \ colon V \ to V}$ ${\ displaystyle T \ colon V \ to V}$ Is such a subspace that $\forall x\in W,T(x)\in W$ ${\ displaystyle \ forall x \ in W, T (x) \ in W}$ ${\ displaystyle \ forall x \ in W, T (x) \ in W}$ , in other words $T(W)\subset W$ ${\ displaystyle T (W) \ subset W}$ ${\ displaystyle T (W) \ subset W}$ .

The invariant subspace is one of the key concepts of linear algebra and functional analysis , which plays an important role in the study of linear mappings acting in finite-dimensional and infinite-dimensional linear spaces .

Examples

Trivial examples are: space itself $V$ ${\ displaystyle V}$ $(W=V)$ ${\ displaystyle (W = V)}$ and zero subspace (consisting of a single zero vector). Invariant subspace $W\subset V$ ${\ displaystyle W \ subset V}$ , $W\neq V$ ${\ displaystyle W \ neq V}$ consisting of more than one zero vector is called an eigenvector .
Linear mapping core $\ker T$ ${\ displaystyle \ ker T}$ .
Important examples of invariant subspaces are eigen and root subspaces of a linear mapping $T$ ${\ displaystyle T}$ .

Literature

Gelfand I. M. Lectures on linear algebra. - 5th. - M .: Dobrosvet, MCLMO , 1998 .-- 319 p. - ISBN 5-7913-0015-8 .
Kostrikin A.I. , Manin Yu. I. Linear algebra and geometry. - M .: Nauka , 1986 .-- 304 p.
Maltsev A. I. Fundamentals of linear algebra. - 3rd. - M .: Nauka , 1970 .-- 400 p.
Shafarevich I.R. , Remizov A.O. Linear algebra and geometry. - M .: Fizmatlit , 2009 .-- 511 p.

Invariant subspace

Examples

Literature

More articles: