Alphabeta MathTriangularisation and Jordan Form: Examples and Counterexamples
About this page
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.
✓ 17 results · all verified · 14 also independently AI-judged
Every result on this page is machine-checked by a proof checker and read in full and owner-audited; the judge is an additional, independent cross-model AI review of the proofs. The 3 not AI-judged were verified by owner audit (typically over a confirmed judge false positive), not failures.
1 · Prerequisites
Abstract Algebra
- Binary Operations, Monoids, Groups and Subgroups
- Cosets, Index and Lagrange's Theorem
- Divisibility, Euclidean Domains, Principal Ideal Domains and Unique Factorisation
- Group Actions, Orbits, Stabilisers and Cayley's Theorem
- Ideals, Quotient Rings and the Isomorphism Theorems for Rings
- Normal Subgroups and Quotient Groups
- Polynomial Rings, the Division Algorithm and Roots
- Rings, Subrings, Integral Domains and Fields
- Simple Field Extensions and the Construction of the Complex Numbers
- Splitting Fields
Linear Algebra
- Determinants of Matrices over a Commutative Ring
- Diagonalisation and the Minimal Polynomial
- Eigenvalues, Eigenvectors and the Characteristic Polynomial
- Linear Independence, Bases and Dimension
- Linear Transformations, Rank-Nullity and Quotient Spaces
- Matrices, the Matrix of a Linear Map, and Change of Basis
- The Determinant of a Linear Operator, Cofactors and Cramer's Rule
- Triangularisation, Generalised Eigenspaces and Jordan Canonical Form
- Vector Spaces, Linear Subspaces, Span and Direct Sums
Real Analysis
2 · Logical flowchart
3 · Items
Example
The quotient of F³ by a coordinate line and its canonical projection
Example
The first isomorphism theorem for (x,y,z)↦(x+y,y+z)
Example
A split 3×3 operator that is triangularisable but not diagonalisable
Example
Building an upper-triangular matrix from a complete invariant flag
Example
Building a Jordan-string basis for a nilpotent operator on F⁶
Example
Recovering J₃(0),J₂(0),J₁(0) from ranks of powers
Example
Recovering a 5×5 Jordan form from shifted power ranks at two eigenvalues
Example
The real quarter-turn acquires diagonal Jordan form over ℂ
False statement
FALSE: Every finite-dimensional endomorphism is triangularisable over its base field
False statement
FALSE: Every finite-dimensional endomorphism has Jordan form over its base field
False statement
FALSE: Equal characteristic and minimal polynomials imply similarity
False statement
FALSE: Jordan canonical form is a unique literal matrix without fixing block order
False statement
FALSE: Geometric multiplicity alone determines Jordan block sizes
Example
Computing commuting diagonal and nilpotent parts of a split Jordan matrix
False statement
FALSE: Every endomorphism has a commuting diagonal-plus-nilpotent decomposition over its base field
Example
A companion operator with a visible cyclic vector and equal canonical polynomials
Counterexample