Linear and multilinear algebra; matrix theory Definition

Nilpotent endomorphism

Definition

Let $F$ be a field, let $V$ be a vector space over $F$ , and let $T \colon V \to V$ be an endomorphism. The map $T$ is nilpotent when $T^k = 0$ for some integer $k \ge 1$ . The least such $k$ is the index of $T$ .

Remarks

If $\dim V = n$ is finite and $T$ is nilpotent, then $T^n = 0$ . In finite dimensions, $T$ is nilpotent if and only if its characteristic polynomial is $t^n$ . Over an algebraically closed field, this means that $0$ is its only eigenvalue. A matrix is nilpotent when its map is nilpotent, see nilpotent matrix. The derivative on the space of polynomials of degree less than $n$ is nilpotent.

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Vector space definition
Linear map definition
Endomorphism definition
Nilpotent endomorphism definition

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Jordan–Chevalley decomposition

Tag	`0M8MHYERTY`
URI	`m2://0M8MHYERTY`
Kind	definition
Writing	W0 stub
MSC	15A04, 15A21
Defines	Nilpotent endomorphism
Uses	Endomorphism

Nilpotent endomorphism

Definition

Remarks

Reading path

Where this leads

Further reading

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