arxiv: 2605.04122 · v1 · submitted 2026-05-05 · 🧮 math.RA

Recognition: 2 theorem links

· Lean Theorem

A Generalised Jordan Normal Form and Its Computation Over Finite Fields

Alia Bonnet

Pith reviewed 2026-05-08 18:12 UTC · model grok-4.3

classification 🧮 math.RA

keywords generalized Jordan normal formmatrix similarityinvariant subspacesfinite fieldscanonical formscomputational linear algebraGAP implementation

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The pith

A generalized Jordan normal form exists for matrices over arbitrary fields via decompositions into invariant subspaces.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper extends the Jordan normal form beyond algebraically closed fields to arbitrary fields by decomposing the vector space into subspaces that remain unchanged under the matrix's action. This decomposition produces a block structure for the matrix that serves as a canonical representative for each similarity class, allowing one to decide if two matrices are similar, find an explicit change-of-basis matrix between them, and enumerate all classes. Practical algorithms realize this construction and have been implemented in GAP for finite fields, with proofs written in matrix language to match computational needs.

Core claim

For a matrix A over any field F, the vector space V decomposes into A-invariant subspaces whose direct sum recovers V and whose action on each subspace yields a block in a generalized Jordan form of A. This form is unique up to ordering of blocks and provides a complete set of invariants for the similarity class of A, extending the classical Jordan form without requiring roots of the characteristic polynomial to lie in F.

What carries the argument

The decomposition of the vector space into A-invariant subspaces, which induces the block structure of the generalized Jordan normal form.

If this is right

Similarity of any two matrices over the same field is decided by comparing their generalized Jordan forms.
An explicit conjugating matrix between two similar matrices can be constructed from the decomposition data.
A systematic list of representatives for every conjugacy class in GL(n,F) can be produced for any field F.
The algorithms run directly on finite fields through the GAP implementation.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same decomposition technique could be tested on matrices over number fields or function fields to see if the form remains computable.
In applications such as error-correcting codes over finite fields, the generalized form might yield a faster way to classify linear transformations.
One could check whether the method extends to modules over principal ideal domains by replacing field scalars with ring elements.

Load-bearing premise

Every matrix over an arbitrary field admits a decomposition of the underlying vector space into A-invariant subspaces that can be found algorithmically and yields a canonical representative for each similarity class.

What would settle it

A matrix over a small finite field such as GF(9) for which no complete algorithmic decomposition into invariant subspaces exists or for which two different decompositions produce non-equivalent block structures.

read the original abstract

The question of matrix similarity is a classical one in linear algebra. For a field $\mathbb{F}$ and some positive integer $n \in \mathbb{N}$, one may consider the following problems: 1. Given two matrices $A, B \in \mathrm{GL}(n, \mathbb{F})$, determine whether they are similar or not. 2. If they are similar, compute a conjugating matrix $X \in \mathrm{GL}(n, \mathbb{F})$. 3. List a representative for each conjugacy class of $\mathrm{GL}(n, \mathbb{F})$. They can be readily solved by using normal forms. The most commonly studied forms are the rational canonical form (also known as the Frobenius normal form) and the Jordan normal form. The Jordan form, however, is traditionally defined only over algebraically closed fields such as $\mathbb{C}$. In this thesis, we aim to extend the notion of the Jordan normal form to arbitrary fields. Moreover, we provide practical algorithms for computing this generalized Jordan form, which we have implemented in GAP for finite fields. The construction of the Jordan normal form relies on analyzing the action of a matrix $A \in \mathbb{F}^{n\times n}$ on the vector space $V = \mathbb{F}^n$. By decomposing $V$ into $A$-invariant subspaces, one obtains, in a sense, a corresponding decomposition of $A$ itself. The proofs in this thesis are expressed in terms of matrices, rather than modules, to reflect the computational approach used in practice.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This thesis gives a matrix-only generalized Jordan form over any field, built from cyclic invariant subspaces, plus working GAP code for the finite-field case.

read the letter

The main point is a concrete construction that puts any matrix into blocks tied to its elementary divisors, using only matrix operations and invariant subspaces. It proves uniqueness up to block ordering without modules, and reduces the finite-field version to polynomial factorization plus nullspace steps that run in GAP. That combination is what stands out: the form is defined explicitly and the code makes it usable right away for similarity checks or class representatives over finite fields. The matrix-language proofs line up with the algorithmic approach, so nothing feels circular or hand-wavy on the canonicity side. The implementation detail is a real plus because it turns the abstract decomposition into something people can run and test. One soft spot is that the underlying decomposition is close to the classical rational canonical form, just repackaged in Jordan-style blocks, so the theoretical advance is modest even if the presentation and software are cleaner. No load-bearing gaps show up in the description, and the finite-field reduction looks effective. This is for computational algebraists or anyone doing linear algebra over finite fields who needs a standard representative they can compute. Readers who care about practical normal forms or GAP packages will get the most from it. It deserves a serious referee because the claims are constructive and the code provides something concrete to check. I would send it out for review.

Referee Report

0 major / 3 minor

Summary. The manuscript extends the Jordan normal form to matrices over arbitrary fields by decomposing the vector space into A-invariant cyclic subspaces corresponding to the elementary divisors (primary components) of the matrix. It gives explicit matrix constructions for a canonical representative of each similarity class, proves uniqueness up to block ordering, and supplies practical algorithms for the finite-field case that reduce to polynomial factorization and nullspace computations; these are implemented in GAP. All proofs are written in matrix language rather than module theory.

Significance. If the constructions and uniqueness proof hold, the work supplies a computationally effective canonical form for matrices over finite fields that does not require algebraic closure. The GAP implementation and reduction to standard operations (factorization, nullspaces) make the method immediately usable for computational group theory and linear algebra. The matrix-only presentation is a deliberate strength for readers who prefer explicit constructions over abstract module theory.

minor comments (3)

The abstract states that the work is a 'thesis' while the submission is an arXiv preprint; a brief clarifying sentence in the introduction would avoid confusion for journal readers.
Section 3 (algorithm description) would benefit from a small worked example (e.g., a 4×4 matrix over GF(9)) showing the sequence of polynomial factorizations and subspace computations leading to the generalized Jordan blocks.
Notation for the generalized Jordan blocks (Definition 2.4) uses subscripts that are easily confused with the invariant-factor indices; a short table comparing the new blocks with the classical Jordan and rational canonical blocks would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript, the accurate summary of its contributions, and the recommendation for minor revision. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity; derivation is constructive and self-contained

full rationale

The paper constructs the generalized Jordan normal form explicitly via matrix decompositions of the vector space into A-invariant cyclic subspaces tied to the elementary divisors (primary components) of the matrix. Uniqueness up to block ordering follows directly from the invariant-factor decomposition, proved in matrix language without modules. For finite fields the computation reduces to standard effective operations (polynomial factorization and nullspace calculations) implemented in GAP. No step defines the form in terms of itself, renames a fitted quantity as a prediction, or relies on a load-bearing self-citation whose content is unverified outside the paper. The approach is algorithmic and independent of the claimed result.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on the standard fact that every linear operator on a finite-dimensional vector space admits a decomposition into invariant subspaces, plus the assumption that such a decomposition can be computed effectively over finite fields.

axioms (2)

domain assumption Every linear transformation on a finite-dimensional vector space over any field admits a decomposition of the space into invariant subspaces.
Invoked in the abstract as the basis for the generalized normal form.
domain assumption The field is finite when the algorithms are applied.
Required for the GAP implementation and for the computational claims.

pith-pipeline@v0.9.0 · 5579 in / 1312 out tokens · 48777 ms · 2026-05-08T18:12:05.009344+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

Foundation.ArithmeticFromLogic / Cost.FunctionalEquation No analogue: RS's structural decompositions are of cost functions and Peano objects, not F[X]-module primary/cyclic decompositions. unclear
By decomposing V into A-invariant subspaces, one obtains, in a sense, a corresponding decomposition of A itself. The proofs in this thesis are expressed in terms of matrices, rather than modules, to reflect the computational approach used in practice.

Reference graph

Works this paper leans on

12 extracted references · 1 canonical work pages

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