Alphabeta MathHow 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.
Invertible square matrices and similarity over a commutative ring
Definition
Let be a commutative ring. A matrix is invertible when there is with . The inverse is unique and is denoted . The general linear group over is
Matrices are similar over when there is such that . Associativity and the identity laws from Matrix arithmetic over a commutative ring is associative, unital and distributive, and transpose reverses products make these the usual units and conjugacy relation in the matrix monoid.
Depends on
- Entrywise ring-matrix operations, rectangular matrix products, identity matrices and transpose
- Matrix arithmetic over a commutative ring is associative, unital and distributive, and transpose reverses products
- Left inverse, right inverse, and invertible element of a monoid
- In a monoid, a left inverse and a right inverse of the same element are equal; hence an invertible element has exactly one inverse, and it is two-sided
Used by
- An invertible square matrix over a commutative ring has unit determinant Corollary
- If A is invertible over a commutative ring, then det(A⁻¹)=det(A)⁻¹ Corollary
- Similar matrices over a commutative ring have the same determinant Corollary
- For n≥ 1, determinant is a natural transformation det:GLₙ(-)⟹(-)^× from commutative rings to groups Example
- For a field, the ring-matrix operations, invertibility and similarity agree exactly with the established field-matrix interface Proposition
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 27 results over 14 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.
Sources
- S. New, MATH 146 Linear Algebra 1 Lecture Notes, Ch. 4 (standard reference, not scraped)