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.
FALSE: Every complex symmetric matrix is unitarily diagonalisable
Statement
Every complex symmetric matrix is unitarily diagonalisable.
Facts & Assumptions
Given: The complex symmetric matrix .
This matrix is symmetric but not normal (A complex symmetric matrix can be nonzero, square to zero, and fail to be normal).
A complex matrix is unitarily diagonalisable exactly when the corresponding operator is normal (Complex spectral theorem: a normal endomorphism of a finite-dimensional complex inner product space has an orthonormal eigenbasis, and conversely).
Refutation
By [L1], the matrix is indeed complex symmetric.
If were unitarily diagonalisable, [L2] would make it normal. That contradicts [L1]. Therefore the claim is false.
Depends on
Used by
Nothing in the library uses this result yet.
Dependency tree · two levels
7 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
- Nicholas Hu, The Schur decomposition (standard reference, not scraped)