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On a nonzero finite-dimensional space, the largest invariant factor is the minimal polynomial

Statement

On a nonzero finite-dimensional space, the largest invariant factor is the minimal polynomial. On the zero space the invariant-factor list is empty and the minimal polynomial is $1$ , so there is no largest invariant factor.

Facts & Assumptions

Given: A nonzero finite-dimensional space with invariant factors $f_{1} ∣ \dots ∣ f_{r}$ supplied by Existence and uniqueness of rational canonical form.

[L1]

The annihilator of $V_{T}$ is generated by the minimal polynomial $μ_{T}$ (For finite-dimensional $V$ , $V_{T}$ is finitely generated and torsion, with annihilator generated by the minimal polynomial).

Proof

technique · direct

1.1

A polynomial $p$ annihilates $⨁_{i} F [x] / (f_{i})$ exactly when $f_{i} ∣ p$ for every $i$ . Since $f_{1} ∣ \dots ∣ f_{r}$ , this is equivalent to $f_{r} ∣ p$ , so the module annihilator is $(f_{r})$ .

givenalgebra

2.1

By [L1], the same annihilator ideal is $(μ_{T})$ . Both generators are monic, so $f_{r} = μ_{T}$ . The nonzero-space hypothesis ensures $r \geq 1$ ; the zero-space convention is as stated.

step 1.1L1∎

Depends on

Used by

Rational canonical form of an explicit four-by-four matrix Example
The polynomial module of a two-by-two operator Example
FALSE: characteristic and minimal polynomials determine similarity False statement
Cayley-Hamilton by the PID-module structure theorem 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

A. Apisa, Wisconsin Math 542, Lemma 34 (standard reference, not scraped)
M. Brussel, Finitely Generated Modules over a PID, Section 5.2 (standard reference, not scraped)