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Jordan canonical form from the elementary divisors of $V_{T}$

Statement

Let $T$ be an endomorphism of a finite-dimensional $F$ -vector space whose characteristic polynomial splits over $F$ . If the elementary divisors of $V_{T}$ are $(x - λ)^{e}$ , then $T$ has a Jordan canonical form with one block $J_{e} (λ)$ for each elementary divisor. The multiset of blocks is unique, and conversely each Jordan block yields its corresponding cyclic primary module. The zero space has empty lists.

Facts & Assumptions

Given: Endomorphism elementary divisors from Invariant factors and elementary divisors of an endomorphism, Jordan canonical form as in Jordan bases and Jordan canonical forms over the base field, and uniqueness of PID elementary divisors (Uniqueness of invariant factors and elementary divisors over a PID).

[L1]

A finitely generated torsion PID module is the direct sum of its prime-power cyclic elementary-divisor summands (Primary decomposition and elementary-divisor form for finitely generated PID modules).

[L2]

$C ((x - λ)^{e})$ is similar to $J_{e} (λ)$ (A companion block for $(x - λ)^{e}$ is similar to the Jordan block $J_{e} (λ)$ ).

[L3]

The product of the invariant factors of $T$ is its characteristic polynomial (The product of the invariant factors is the characteristic polynomial).

Proof

technique · constructive

1.1

By [L3], every invariant factor divides the split polynomial $χ_{T}$ , so every irreducible factor occurring in an elementary divisor is linear. By [L1], $V_{T}$ is the direct sum of cyclic modules $F [x] / ((x - λ)^{e})$ , one for each elementary divisor.

L1L3given

2.1

On each cyclic summand, multiplication by $x$ has companion matrix $C ((x - λ)^{e})$ , which [L2] turns by a basis change into $J_{e} (λ)$ . Concatenating these bases constructs a Jordan basis for $T$ .

step 1.1L2construct

3.1

Every Jordan block gives the reverse cyclic module $F [x] / ((x - λ)^{e})$ , and uniqueness of elementary divisors gives uniqueness of the block multiset up to order. Empty elementary-divisor data gives the empty Jordan form on the zero space.

step 2.1givendischarge-construct∎

Remarks

This proof obtains the blocks from the $F [x]$ -module structure. The published criterion Jordan form over the base field exists exactly when the characteristic polynomial splits obtains existence through generalized eigenspaces and Jordan strings.

Depends on

Used by

Dependency tree · two levels

23 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, Corollary 38 (standard reference, not scraped)
M. Brussel, Finitely Generated Modules over a PID, Section 7.4 (standard reference, not scraped)

Jordan canonical form from the elementary divisors of VTVT​