DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)judge pass (deepseek-v4-pro + claude-opus-5[1m])audited 2026-08-24

How 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.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

The companion matrix of a monic polynomial

Definition

For a monic polynomial

$f (x) = x^{d} + a_{d - 1} x^{d - 1} + \dots + a_{1} x + a_{0}$

of positive degree over a commutative ring, its companion matrix is

$C (f) = (\begin{array}{ccccc} 0 & 0 & \dots & 0 & - a_{0} \\ 1 & 0 & \dots & 0 & - a_{1} \\ 0 & 1 & ⋱ & 0 & - a_{2} \\ ⋮ & ⋱ & ⋱ & ⋮ & ⋮ \\ 0 & \dots & 0 & 1 & - a_{d - 1} \end{array}) .$

Thus the ones are on the subdiagonal. This is the matrix of multiplication by $x$ on the power basis $1, x, \dots, x^{d - 1}$ of the cyclic module $R [x] / (f)$ .

Depends on

Used by

Dependency tree · two levels

6 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

M. Brussel, Finitely Generated Modules over a PID, Section 7 (standard reference, not scraped)
A. Apisa, Wisconsin Math 542, Lecture 11 (standard reference, not scraped)