Linear and multilinear algebra; matrix theory Definition

Semisimple endomorphism

Definition

Let $F$ be a field, let $V$ be a finite-dimensional vector space over $F$ , and let $T \colon V \to V$ be an endomorphism. The map $T$ is semisimple when, for each $T$ -invariant subspace $W$ of $V$ , there is a $T$ -invariant subspace $W'$ with $V = W \oplus W'$ .

Remarks

The map $T$ is semisimple if and only if its minimal polynomial has no repeated irreducible factor. Over an algebraically closed field, $T$ is semisimple if and only if it is diagonalisable. The rotation of $\R^2$ by $90$ degrees is semisimple, but it is not diagonalisable over $\R$ .

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

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Jordan–Chevalley decomposition

Tag	`0QS8P74F2Z`
URI	`m2://0QS8P74F2Z`
Kind	definition
Writing	W0 stub
MSC	15A04, 15A20
Defines	Semisimple endomorphism
Uses	Invariant subspace

Semisimple endomorphism

Definition

Remarks

Reading path

Where this leads

Further reading

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