LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-6-luna)

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.

Finite matrix trace cyclicity and nondegeneracy

Statement

Let $d, e \geq 1$ . For $A \in M_{e \times d} (C)$ and $B \in M_{d \times e} (C)$ , ${Tr}_{e} (A B) = {Tr}_{d} (B A),$ where ${Tr}_{r}$ is the diagonal-sum trace on $M_{r} (C)$ . This includes cyclicity for square matrices. For $C \in M_{d} (C)$ , the bilinear trace pairing is nondegenerate: if ${Tr}_{d} (X C) = 0$ for every $X \in M_{d} (C)$ , then $C = 0$ . No adjoint is inserted in this pairing.

Facts & Assumptions

Given: Integers $d, e \geq 1$ , rectangular matrices $A \in M_{e \times d} (C)$ , $B \in M_{d \times e} (C)$ , a matrix $C \in M_{d} (C)$ , and standard matrix units $E_{i j}$ on $C^{d}$ .

Proof

1.1givenalgebra

By matrix multiplication and the diagonal-sum trace, ${Tr}_{e} (A B) = \sum_{i = 1}^{e} \sum_{j = 1}^{d} A_{i j} B_{j i}$ . These are finite scalar sums, so reordering them gives $\sum_{j = 1}^{d} \sum_{i = 1}^{e} B_{j i} A_{i j} = {Tr}_{d} (B A)$ . Taking $e = d$ gives square trace cyclicity.

2.1givenalgebra∎

If ${Tr}_{d} (X C) = 0$ for all $X$ , choose $X = E_{i j}$ . Direct multiplication gives ${Tr}_{d} (E_{i j} C) = C_{j i}$ , so $C_{j i} = 0$ for every $i, j$ . Thus every matrix entry of $C$ vanishes and $C = 0$ .

Dependency tree · 0 levels

Nothing. This result depends on no other item in the library.

Finite matrix trace cyclicity and nondegeneracy

Statement

Facts & Assumptions

Proof

Dependency tree · 0 levels

Sources