DefinitionDefinition: AI-adaptedProof: Not applicableaudited 2026-08-13

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.

For $A \in M_{n} (F)$ , the characteristic polynomial is $χ_{A} (x) = \det (x I_{n} - A)$ when $n \geq 1$ , with $χ_{A} (x) = 1$ for the unique $0 \times 0$ matrix

Definition

Let $F$ be a field and let $A \in M_{n} (F)$ . If $n \geq 1$ , regard $x I_{n} - A$ as a matrix over the commutative polynomial ring $F [x]$ . The characteristic polynomial of $A$ is

$χ_{A} (x) : = \det (x I_{n} - A) \in F [x] .$

For $n = 0$ , define the characteristic polynomial of the unique $0 \times 0$ matrix to be the constant polynomial $1$ . This agrees with the empty-product convention and makes the characteristic polynomial monic of degree $0$ in the zero-sized case.

For $A \in M_{n} (F)$ , the characteristic polynomial is $χ_{A} (x) = \det (x I_{n} - A)$ when $n \geq 1$ , with $χ_{A} (x) = 1$ for the unique $0 \times 0$ matrix

Definition

Depends on

Dependency tree · 2 levels

Sources

For A∈Mn(F)A∈Mn​(F), the characteristic polynomial is χA(x)=det⁡(xIn−A)χA​(x)=det(xIn​−A) when n≥1n≥1, with χA(x)=1χA​(x)=1 for the unique 0×00×0 matrix

Definition

Depends on

Dependency tree · 2 levels

Sources

For $A \in M_{n} (F)$ , the characteristic polynomial is $χ_{A} (x) = \det (x I_{n} - A)$ when $n \geq 1$ , with $χ_{A} (x) = 1$ for the unique $0 \times 0$ matrix