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Invariant factors and elementary divisors of an endomorphism
Definition
Let be an endomorphism of a finite-dimensional vector space over . The invariant factors and elementary divisors of are those of the -module (The -module of an endomorphism), normalized to be monic polynomials.
The ring is a PID by For every field , is a principal ideal domain, and For finite-dimensional , is finitely generated and torsion, with annihilator generated by the minimal polynomial makes a finitely generated torsion module. These data therefore exist by Invariant-factor decomposition of a finitely generated module over a PID and Primary decomposition and elementary-divisor form for finitely generated PID modules, and are unique by Uniqueness of invariant factors and elementary divisors over a PID.
Depends on
- Invariant factors and elementary divisors of a finitely generated module over a PID
- The $F[x]$-module $V_T$ of an endomorphism
- For every field $F$, $F[x]$ is a principal ideal domain
- For finite-dimensional $V$, $V_T$ is finitely generated and torsion, with annihilator generated by the minimal polynomial
- Invariant-factor decomposition of a finitely generated module over a PID
- Primary decomposition and elementary-divisor form for finitely generated PID modules
- Uniqueness of invariant factors and elementary divisors over a PID
Used by
Dependency tree · two levels
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Sources
- M. Brussel, Finitely Generated Modules over a PID, Sections 5.2-5.3 (standard reference, not scraped)
- A. Apisa, Wisconsin Math 542, Lecture 11 (standard reference, not scraped)