Alphabeta MathHow statement and proof provenance work
The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
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Nilpotent endomorphisms and their nilpotency index
Definition
An endomorphism is nilpotent if for some positive integer . Its nilpotency index is the least positive integer such that ; this least exponent exists by the well-ordering principle (The well-ordering principle).
The unique endomorphism of the zero space is nilpotent. With the same least-positive-exponent convention, its nilpotency index is because its identity and zero endomorphisms coincide.
Depends on
Used by
- Nilpotent transformations and nil representations Definition
- Semisimple endomorphisms as endomorphisms diagonalisable over an algebraic closure, and nilpotent endomorphisms Definition
- FALSE: Every endomorphism has a commuting diagonal-plus-nilpotent decomposition over its base field False statement
- Characterisations of a nilpotent endomorphism Theorem
- Every finite-dimensional nilpotent endomorphism has a basis of Jordan strings Theorem
Dependency tree · two levels
12 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- S. Axler, Linear Algebra Done Right, 4th ed., Section 8B (standard reference, not scraped)