CorollaryStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-08-11

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.

Similar matrices have the same trace

If square matrices $A$ and $B$ over $F$ are similar, then $tr (A) = tr (B)$ .

Given: Similar matrices $A, B \in M_{n} (F)$ .

[L1]

Similarity means $B = P^{- 1} A P$ for an invertible matrix $P$ (Similar matrices: $B = P^{- 1} A P$ for an invertible $P$ ).

[L2]

[L3]

technique · direct

1.1givenL1

Choose invertible $P$ with $B = P^{- 1} A P$ as in [L1].

2.1step 1.1L1L2

Apply [L2] to $X = P^{- 1} A$ and $Y = P$ to obtain $tr (B) = tr (P (P^{- 1} A))$ .

3.1step 2.1L1L2L3∎

By associativity and unitality from [L3], $P (P^{- 1} A) = (P P^{- 1}) A = I_{n} A = A$ , so $tr (B) = tr (A)$ .

Depends on