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.
Similar matrices: for an invertible
Definition
Two matrices are similar when there is an invertible matrix such that
The matrix is a change-of-basis matrix or conjugating matrix for this similarity. Replacing by its inverse exchanges the roles of and .
Depends on
Used by
- Similar matrices have the same trace Corollary
- A nonzero square-zero matrix is not similar to any diagonal matrix Example
- For a field, the ring-matrix operations, invertibility and similarity agree exactly with the established field-matrix interface Proposition
- Similar matrices have the same characteristic polynomial Theorem
- Similarity is an equivalence relation, and two matrices represent the same endomorphism in two bases exactly when they are similar Theorem
- Two matrices are similar exactly when their invariant factors agree Theorem
Dependency tree · two levels
3 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. Schiavone, MIT 18.700 Day 9, Definition 38 (standard reference, not scraped)