Linear and multilinear algebra; matrix theory Definition

Restriction of an endomorphism to an 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. Let $W$ be a $T$ -invariant subspace of $V$ . The restriction of $T$ to $W$ is the endomorphism $T|_W \colon W \to W, \qquad T|_W(w) = T(w).$

Remarks

The map is well defined because $T(W) \subseteq W$ , and it is linear. If $V$ is finite-dimensional, then the characteristic polynomial of $T|_W$ divides that of $T$ . Take a basis of $V$ that extends a basis of $W$ . In this basis, the matrix of $T$ is block upper triangular, and its first diagonal block is the matrix of $T|_W$ .

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Vector space definition
Linear map definition
Linear subspace definition
Endomorphism definition
Invariant subspace definition
Restriction of an endomorphism to an invariant subspace definition

Tag	`0PFPWKBDTJ`
URI	`m2://0PFPWKBDTJ`
Kind	definition
Writing	W0 stub
MSC	15A04, 47A15
Defines	Restriction of an endomorphism to an invariant subspace
Uses	Invariant subspace

Restriction of an endomorphism to an invariant subspace

Definition

Remarks

Reading path

Further reading

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