Linear and multilinear algebra; matrix theory Definition

Invariant subspace

Definition

Let $F$ be a field, let $V$ be a vector space over $F$ , and let $T \colon V \to V$ be an endomorphism. A subspace $W$ of $V$ is invariant under $T$ , or $T$ -invariant, when $T(w) \in W$ for all $w \in W$ .

Remarks

The subspaces $0$ and $V$ , $\ker T$ and $\im T$ , and each eigenspace of $T$ are $T$ -invariant. A line spanned by $v \ne 0$ is $T$ -invariant if and only if $v$ is an eigenvector of $T$ . If $V = W_1 \oplus W_2$ with both $W_i$ invariant, then the matrix of $T$ in an adapted basis is block diagonal.

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Vector space definition
Linear map definition
Linear subspace definition
Endomorphism definition
Invariant subspace definition

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Tag	`0EP59ESQ94`
URI	`m2://0EP59ESQ94`
Kind	definition
Writing	W1 stated
MSC	15A04, 47A15
Defines	Invariant subspace
Uses	Endomorphism, Linear subspace

Invariant subspace

Definition

Remarks

Reading path

Where this leads

Further reading

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