Alphabeta MathDefinitionDefinition: Literature-sourcedProof: Not applicableaudited 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.
The trace as the sum of the diagonal entries
Definition
For a square matrix , its trace is the sum of its diagonal entries,
For , this is the empty sum, so the unique empty matrix has trace .
Depends on
Used by
- For an invertible matrix over a field, Cayley-Hamilton makes every matrix-power entry and trace sequence linearly recurrent Corollary
- Congruence need not preserve trace or determinant: the real 1×1 matrices [1] and [4] are congruent Counterexample
- χ_A(x) is monic of degree n; for n≥1 its xⁿ⁻¹ coefficient is -tr(A) and its constant coefficient is (-1)ⁿ det(A), while χ_0×0=1 Lemma
- For matrices over a field, the commutative-ring trace agrees with the published matrix trace Proposition
- Trace and Frobenius-linear matrix functionals differentiate by inspection Proposition
- Trace is a linear functional on Mₙ(F) Proposition
- For A∈ M_m× n(F) and B∈ M_n× m(F), tr(AB)=tr(BA) Theorem
- The determinant differential is D det(A)[H]=tr(adj(A)H) at every matrix, and Jacobi's formula holds on the invertible locus Theorem
Dependency tree · two levels
13 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
- S. Axler, Linear Algebra Done Right, 4th ed., Definition 8.47 (standard reference, not scraped)