An Alternative Framework for Irreducibility and Primitivity of Nonnegative Tensors

Jianhong Xu

An Alternative Framework for Irreducibility and Primitivity of Nonnegative Tensors

T0 review · 0 major / 7 minor · reviewed 2026-08-02 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read A nonnegative tensor is s-primitive exactly when it is s-irreducible, uniformly accessible on arrows, and aperiodic at every index—a graph-theoretic test for a new kind of tensor primitivity.

desk verdict Solid and self-contained extension of the matrix primitivity theorem to nonnegative tensors; Theorem 2.5 is proved correctly, with only minor deferred proofs from the author's own earlier papers. read the letter →

arxiv 2606.30482 v2 pith:O65FFM25 submitted 2026-06-29 math.RA

Jianhong Xu This is my paper

classification math.RA MSC 15A6915A7215B4846B28

keywords nonnegativetensorstensorboxproducts-irreducibilitys-primitivityaperiodicityaccessibilityhigher-orderMarkovchainsprimitivitycharacterization

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper tries to establish a graph-theoretic characterization of a new kind of primitivity for nonnegative tensors, called s-primitivity, defined through a tensor power built from the non-associative box product. The central theorem states that a tensor is s-primitive if and only if it is s-irreducible, every accessible pair has a nonempty uniform accessibility set S_A(i,j), and every index has period 1. For ordinary matrices (order 2), the characterization reduces to the classical result that a nonnegative matrix is primitive iff it is irreducible and aperiodic. This matters because the new notions translate directly into ergodicity and regularity for higher-order Markov chains, where the classical matrix theory does not apply.

What carries the argument

The load-bearing object is the box product A⊠B and its recursively defined right power A^α = A^(α−1) ⊠ A, with identity tensor I that is only a left identity. Because the box product is not associative for m ≥ 3, this right-recursive power is a convention, not a canonical operation. The reduced matrix Q_A—an n^{m−1}×n^{m−1} matrix built from the tensor's entries—is what makes the framework tractable: Theorem 2.3 shows A^(α+β) = A^(α) Q_A^β, so tensor power entries are governed by ordinary matrix powers of Q_A. The sets S_A(i,j) of uniform return/access exponents, together with the gcd period d = gcd(S_A(i)), supply the combinatorial control for the characterization.

What would settle it

Take the finite collection of all 3rd-order 3-dimensional tensors with entries in {0,1}; for each tensor, compute the right-recursive powers A^α until the zero-nonzero pattern repeats (which must happen, since there are finitely many patterns), and compare the predicate 'some A^α has all entries positive' with the conjunction of conditions (i)-(iii) from Theorem 2.5. One tensor where the two answers disagree would refute the characterization; an exhaustive match would corroborate it.

Watch

Extended reading notes

Core claim

On the paper's own terms, the discovery is Theorem 2.5: for an mth order, n-dimensional nonnegative tensor A, s-primitivity—meaning some power A^α built by repeated right box products has all entries positive—is equivalent to three graph-theoretic conditions: (i) A is s-irreducible, i.e., every entry position of some power is positive with an exponent that may depend on the position; (ii) for every arrow i→j in the accessibility digraph, the set S_A(i,j) of exponents that simultaneously work for all intermediate indices is nonempty; and (iii) every index i has gcd(S_A(i)) = 1. The proof works by showing through the reduced matrix Q_A that tensor power entries follow matrix-power dynamics, th

Load-bearing premise

The framework stands or falls on the convention that A^alpha means repeated multiplication on the right, A^alpha = A^(alpha-1) ⊠ A; the paper itself notes after Definition 2.1 that the box product is not associative, so this bracketing is a choice, and changing it would change which tensors count as s-irreducible or s-primitive.

Editorial extensions

If this is right

For m=2, Theorem 2.5 restores the classical theorem: a nonnegative matrix is primitive iff it is irreducible and every index is aperiodic.
S-primitivity and the existing tensor primitivity are logically independent in general: the paper gives examples of a primitive but not s-irreducible tensor, an s-primitive but not primitive tensor, and an s-irreducible and primitive but not s-primitive tensor.
S-irreducibility implies the standard tensor irreducibility, but not conversely; so the new framework is strictly finer at the irreducible level.
If A is s-irreducible, then c1I + c2A is s-primitive for any c1,c2>0; in particular, positive diagonal entries make an s-irreducible tensor s-primitive.
In higher-order Markov chains, s-primitivity is regularity, so Theorem 2.5 provides a checkable necessary-and-sufficient route to the existence of a limiting distribution, and Theorem 2.7 gives the simple sufficient condition that an ergodic chain with positive diagonal transition entries is regular.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Because Theorem 2.3 reduces tensor powers to powers of Q_A, questions like the minimal α with A^α > 0—or a bound analogous to the classical primitive-matrix exponent bound—can likely be answered through the spectral and graph theory of Q_A, which is an ordinary n^{m−1}×n^{m−1} matrix.
The right-recursive power convention is one of several possible bracketings of the non-associative box product; defining A^α by left multiplication or another bracketing would produce different s-notions, and comparing them could reveal which convention truly matches the transition structure of higher-order Markov chains.
Condition (ii) can be read as uniform accessibility: whenever j is reachable from i, there is one step count that works no matter which intermediate indices appear. One could test on real or simulated transition tensors whether ergodic chains that violate this uniformity are exactly the ones whose higher-order Markov chains show slow mixing or no regular limit.
A finite exhaustive check is feasible: for all 0-1 tensors of order 3 and dimension 3 (2^27 ≈ 1.3×10^8 cases), compare s-primitivity with conditions (i)-(iii); any mismatch would refute Theorem 2.5.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The main thing to know: this is a legitimate, self-contained extension of the classical matrix primitivity theorem to m-th order nonnegative tensors via a specific non-associative "box" product and a right-recursive power. The central result, Theorem 2.5, states that s-primitivity is equivalent to s-irreducibility plus nonempty exponent sets on all accessible pairs plus aperiodicity of every vertex. The proof is written out in detail and I checked the steps: the necessity is immediate, and the sufficiency uses Lemma 2.2 and Lemma 2.3 with a finite-subset gcd argument that holds because the gcd sequence over an enumeration of a subset stabilizes to the gcd of the whole set. The non-associativity of the box product is explicitly acknowledged and not a hidden assumption; the power is defined recursively on the right, and the proof never uses associativity. In the matrix case m=2 condition (ii) becomes vacuous and the theorem reduces to the classical result. That is a real contribution.

What the paper does well: it gives a uniform algebraic treatment that includes the earlier stochastic-third-order results and the m=2 case, and it provides several examples showing how s-primitivity and the existing primitivity notion differ. The mode-1 matricization and reduced matrix setup is helpful, and Theorem 2.3 is proved cleanly with matrix multiplication.

Soft spots, in proportion: two supporting results, Theorem 2.1 and Theorem 2.4, are stated with proofs "easily extended" from the author's own stochastic-tensor papers. The stress-test note already shows Theorem 2.4 follows directly from Theorem 2.3, and Theorem 2.1 follows from a cut argument — so these are genuine but minor gaps. A referee would want the details supplied, not a rewrite. The framework does depend on the right-recursive power convention; a different bracketing of the box product would change the definitions and the exponent sets. That is not a flaw in the math, only a reminder that the notions are convention-dependent. The citation pattern is heavily self-referential, but the cited results are either verified in the text or easy to expand, so I don't see that as a problem.

Who this is for: anyone working on nonnegative tensor theory, higher-order Markov chains, or generalizations of matrix primitivity. It deserves a serious referee; I would have no trouble sending it out. The central theorem is proved correctly and the presentation is concise.

Recommendation: send to peer review.

Referee Report

0 major / 7 minor

Summary. The paper proposes an alternative framework for irreducibility and primitivity of nonnegative tensors, based on the previously introduced 'box' product and its recursively defined right power. It defines s-irreducibility and s-primitivity (Definitions 2.5 and 2.6), explores their relationship with the classical tensor notions (Theorems 2.1, 2.2, 2.4, and examples), and proves a main characterization theorem (Theorem 2.5): a nonnegative tensor is s-primitive iff it is s-irreducible, its uniform accessibility sets S_A(i,j) are nonempty whenever i→j, and all indices are aperiodic. This reduces to the classical matrix characterization when m=2. The paper concludes with auxiliary results (Theorems 2.6 and 2.7) and a discussion of applications to higher-order Markov chains.

Significance. The main result, Theorem 2.5, is a clean and potentially useful generalization of the classical Frobenius characterization for nonnegative matrices. The proof is self-contained and, on inspection, correct: Lemma 2.2 is proved through Theorem 2.3, the finite-subset gcd argument is valid, and the numerical semigroup lemma is applied appropriately. The paper gives explicit examples that distinguish the new notions from existing tensor primitivity and shows the reduction to the matrix case. The framework is a modeling choice because the box product is non-associative, but this is explicitly acknowledged and the right-recursive power is well-motivated by the higher-order Markov chain interpretation. The contribution is modest but solid; it broadens previous third-order stochastic-tensor results to arbitrary order and to general nonnegative tensors.

minor comments (7)

[§2, Theorems 2.1 and 2.4] The proofs of Theorems 2.1 and 2.4 are deferred to the author's own papers with the remark that the arguments are easily extended. Since Theorem 2.3 is proved, Theorem 2.4 follows immediately by A^{(α)} = A^{(0)} Q_A^α; and Theorem 2.1 follows by a simple induction on α. Please include these short proofs to make the paper self-contained.
[Definition 2.8] The description of the mode-1 matricization is ambiguous: it says 'linear indexing order of i_3,...,i_m', but the multi-index column is i_2 i_3 ... i_m. Please clarify the exact ordering of columns.
[Definition 2.7] There is a typo in the line 'for any i1i1 . . . , im−1' — presumably this should be 'i1i2 . . . im−1'. Please correct.
[Theorem 2.5, necessity part] The necessity of conditions (ii) and (iii) is stated as 'straightforward'. For (ii), it would be helpful to note explicitly that s-primitivity gives A^α > 0 for all sufficiently large α, so that a single exponent belongs to S_A(i,j). A one-line justification would improve clarity.
[Theorem 2.6, proof] The proof of the inequality (I+A)^β ≥ Σ_{α=0}^β A^α is only sketched for β=2,3 and then asserted for general β. Please provide a formal induction, using the monotonicity of the box product and the fact that A^α ⊠ I ≥ 0.
[Theorem 2.2] The wording 'both s-primitivity and primitivity' should be 'both s-primitive and primitive'. Similar grammatical issues appear in a few places.
[Example 2.2] The claim 'A^α = A for all α≥2' is asserted without verification. A short explanation would be useful, since the box product is not associative.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: Theorem 2.5 is proved from explicit definitions and independent lemmas.

full rationale

The paper's central characterization, Theorem 2.5, is derived independently inside the text. Its sufficiency proof uses Lemma 2.2 (proved in the paper from Theorem 2.3, which carries a full algebraic proof using Definitions 2.1, 2.2, 2.7 and 2.8) and Lemma 2.3 (an external numerical-semigroup fact). The finite gcd stabilization argument is valid. The necessity direction checks that s-primitivity implies s-irreducibility, the S_A(i,j) conditions, and aperiodicity; it does not assume Theorem 2.5. No fitted parameter is renamed as a prediction, and the s-framework is not defined in terms of Theorem 2.5. Self-citations appear, but they are not load-bearing: Theorem 2.1 is cited to [7] and Theorem 2.4 is cited to [10], both with 'easily extended' notes, yet neither is used in the proof of Theorem 2.5; Theorem 2.3 — the key identity A^(α+β)=A^(α)Q_A^β — is proved here. The non-associativity of the box product is explicitly stated, and the recursive right-power convention in Definition 2.2 is a stated modeling choice rather than a hidden input. Deferred trivial proofs (Theorem 2.2, Lemma 2.4) and the MATLAB implementation reference [28] do not affect the central claim. Therefore there is no circular step: the central claim does not reduce by construction to its inputs.

Assumptions & free parameters 0 free parameters · 6 assumptions · 4 invented entities

No numerical fitting; this is a pure mathematics paper. The structural choices that matter are the right-recursive box-product power convention, the identity tensor, the reduced-matrix matricization, and the stochastic-to-nonnegative extensions; these are listed as axioms rather than numerical free parameters.

assumptions (6)

standard math Classical matrix irreducibility/primitivity and their graph/pattern characterizations are as stated in Definitions 1.1-1.3.
Invoked throughout; cited [2,11].
domain assumption The existing tensor notions of irreducibility (Definition 1.4) and primitivity (Definition 1.6) are the accepted baseline.
Paper positions its new definitions relative to these; cited [18,3].
ad hoc to paper The box product and its right-recursive power are well-defined and the natural generalization of matrix multiplication to tensors.
Definitions 2.1-2.2; non-associativity makes this a convention rather than canonical. Underlies all results.
ad hoc to paper Results in [7, Theorem 3.2] and [10, Theorem 3.1] proved for stochastic tensors remain true for all nonnegative tensors.
Used in Theorems 2.1 and 2.4 proofs; extension asserted without proof.
domain assumption Accessibility relation and the equivalence relation defined in [9,24] extend to m-th order nonnegative tensors.
Definition 2.9 and used in Theorem 2.5.
standard math The numerical semigroup result (Lemma 2.3) from [1] holds for finite sets of positive integers with gcd 1.
Used in Theorem 2.5 to obtain a uniform beta_j.

invented entities (4)

s-irreducibility and s-primitivity
purpose: Alternative definitions of tensor irreducibility and primitivity based on box-product powers.
New concepts introduced in this paper; no external falsifiable handle.
Identity tensor I with entries delta_{i1 i2}
purpose: Left identity for the box product; used in Theorem 2.3 and Theorem 2.6.
Defined by convention; not a physical entity and not externally testable.
Reduced matrix Q_A
purpose: Matricization enabling A^(alpha+beta) = A^(alpha) Q_A^beta.
Adapted from the author's earlier work [10]; a construction, not an entity with independent evidence.
Exponent sets S_A(i,j)
purpose: Record exponents for which selected tensor-power entries are positive; central to Theorem 2.5.
New device introduced in this paper; definitional rather than empirical.

how reviews work

0 comments

Cite this review

Pith. "Pith review of An Alternative Framework for Irreducibility and Primitivity of Nonnegative Tensors." pith.science (2026). https://pith.science/paper/O65FFM25

@misc{pith2026260630482,
  author       = {Pith},
  title        = {Pith review of: An Alternative Framework for Irreducibility and Primitivity of Nonnegative Tensors},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/O65FFM25}},
  note         = {Machine review of arXiv:2606.30482}
}

read the original abstract

Motivated by some recent studies on higher order Markov chains and well-known characterizations for irreducibility and primitivity of nonnegative matrices, we propose in this paper an alternative framework for irreducibility and primitivity of nonnegative tensors, giving rise to the concepts of s-irreducibility and s-primitivity. This framework includes the relevant results on matrices as its special cases, yet it expands existing results regarding irreducibility and primitivity for tensors. In addition to its tensor theoretic significance, such a framework has important implications for applied fields, especially when it comes to higher order Markov chains.

Review history (2 revisions) →

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

28 extracted references · 1 linked inside Pith

[1]

Bapat, T

R. Bapat, T. Raghavan,Nonnegative Matrices and Applications, Cam- bridge University Press, 1997

1997
[2]

Berman, R

A. Berman, R. Plemmons,Nonnegative Matrices in the Mathematical Sciences, SIAM, 1994

1994
[3]

Chang, K

K. Chang, K. Pearson, T. Zhang, Primitivity, the convergence of the NQZ method, and the largest eigenvalue for nonnegative tensors,SIAM Journal on Matrix Analysis & Applications32: 806–819, 2011

2011
[4]

Chang, T

K. Chang, T. Zhang, On the uniqueness and non-uniqueness of the pos- itiveZ-eigenvector for transition probability tensors,Journal of Mathe- matical Analysis & Applications408: 525–540, 2013. 18

2013
[5]

L. Cui, W. Li, M. Ng, Primitive tensors and directed hypergraphs,Linear Algebra & Its Applications471: 96–108, 2015

2015
[6]

Gleich, L

D. Gleich, L. Lim, Y. Yu, Multilinear pagerank,SIAM Journal on Matrix Analysis & Applications36: 1507–1541, 2015

2015
[7]

L. Han, K. Wang, J. Xu, Higher order ergodic Markov chains and first passage times,Linear & Multilinear Algebra70: 6772–6779, 2022

2022
[8]

L. Han, J. Xu, Ever-reaching probabilities and mean first passage times of higher order ergodic Markov chains,Linear & Multilinear Algebra72: 59–75, 2024

2024

Show all 28 references

[9]

L. Han, J. Xu, On classification of states in higher order Markov chains, Linear Algebra & Its Applications685: 24–45, 2024

2024
[10]

L. Han, J. Xu, On limiting probability distributions of higher order Markov chains,Linear & Multilinear Algebra74: 740–756, 2026

2026
[11]

R. Horn, C. Johnson,Matrix Analysis, Cambridge University Press, 1985

1985
[12]

S. Hu, L. Qi, Convergence of a second order Markov chain,Applied Mathematics & Computation241: 183–192, 2014

2014
[13]

Iosifescu,Finite Markov Processes & Their Applications, Dover Pub- lications, 2007

M. Iosifescu,Finite Markov Processes & Their Applications, Dover Pub- lications, 2007

2007
[14]

Kemeny, J

J. Kemeny, J. Snell,Finite Markov Chains, Springer-Verlag, 1960

1960
[15]

Kolda, B

T. Kolda, B. Bader, Tensor decompositions and applications,SIAM Review51: 455–500, 2009

2009
[16]

C. Li, S. Zhang, Stationary probability vectors of higher-order Markov chains,Linear Algebra & Its Applications473: 114–125, 2016

2016
[17]

W. Li, M. Ng, On the limiting probability distribution of a transition probability tensor,Linear Algebra & Its Applications62: 362–385, 2014

2014
[18]

L. Lim, Singular values and eigenvalues of tensors: a variational ap- proach,1st IEEE International Workshop on Computational Advances in Multi-Sensor Adaptive Processing: 129–132, 2005. 19

2005
[19]

Martin, R

C. Martin, R. Shafer, B. Larue, An order-ptensor factorization with applications in imaging,SIAM Journal on Scientific Computing35: 474– 490, 2013

2013
[20]

L. Qi, Z. Luo,Tensor Analysis: Spectral Theory & Special Tensors, SIAM, 2017

2017
[21]

Rosales, P

J. Rosales, P. Garc ´ ıa-S´ anchez,Numerical Semigroups, Springer, 2009

2009
[22]

Smilde, R

A. Smilde, R. Bro, P. Geladi,Multi-Way Analysis: Applications in the Chemical Sciences, Wiley, 2004

2004
[23]

Vladimirescu, Periodicitate in lanturile Markov duble omogene,Studii si Cercetari de Matematica36: 559–561, 1984

I. Vladimirescu, Periodicitate in lanturile Markov duble omogene,Studii si Cercetari de Matematica36: 559–561, 1984

1984
[24]

Vladimirescu, Lanturi Markov duble omogene regulate,Analele Uni- versitatii Din Craiova, Seria Matematica, Fizica-Chimie13: 59–62, 1985

I. Vladimirescu, Lanturi Markov duble omogene regulate,Analele Uni- versitatii Din Craiova, Seria Matematica, Fizica-Chimie13: 59–62, 1985

1985
[25]

S. Wu, M. Chu, Markov chains with memory, tensor formulation, and the dynamics of power iteration,Applied Mathematics & Computation303: 226–239, 2017

2017
[26]

Xu, Can a higher order Markov chain be treated as a first order Markov chain?,Probability in the Engineering & Informational Sciences 40: 400–415, 2026

J. Xu, Can a higher order Markov chain be treated as a first order Markov chain?,Probability in the Engineering & Informational Sciences 40: 400–415, 2026

2026
[27]

Xu, On computations of limiting probability distributions of higher order Markov chains,Applied Mathematics & Computation531: 130189, 2026

J. Xu, On computations of limiting probability distributions of higher order Markov chains,Applied Mathematics & Computation531: 130189, 2026

2026
[28]

Xu, HOMC: a MATLAB package for higher order Markov chains, under review,https://doi.org/10.48550/arXiv.2510.02664

J. Xu, HOMC: a MATLAB package for higher order Markov chains, under review,https://doi.org/10.48550/arXiv.2510.02664. 20

doi arXiv

Pith tools

Reviewed August 2, 2026 · model on record in the stance chip above.

[1] [1]

Bapat, T

R. Bapat, T. Raghavan,Nonnegative Matrices and Applications, Cam- bridge University Press, 1997

1997

[2] [2]

Berman, R

A. Berman, R. Plemmons,Nonnegative Matrices in the Mathematical Sciences, SIAM, 1994

1994

[3] [3]

Chang, K

K. Chang, K. Pearson, T. Zhang, Primitivity, the convergence of the NQZ method, and the largest eigenvalue for nonnegative tensors,SIAM Journal on Matrix Analysis & Applications32: 806–819, 2011

2011

[4] [4]

Chang, T

K. Chang, T. Zhang, On the uniqueness and non-uniqueness of the pos- itiveZ-eigenvector for transition probability tensors,Journal of Mathe- matical Analysis & Applications408: 525–540, 2013. 18

2013

[5] [5]

L. Cui, W. Li, M. Ng, Primitive tensors and directed hypergraphs,Linear Algebra & Its Applications471: 96–108, 2015

2015

[6] [6]

Gleich, L

D. Gleich, L. Lim, Y. Yu, Multilinear pagerank,SIAM Journal on Matrix Analysis & Applications36: 1507–1541, 2015

2015

[7] [7]

L. Han, K. Wang, J. Xu, Higher order ergodic Markov chains and first passage times,Linear & Multilinear Algebra70: 6772–6779, 2022

2022

[8] [8]

L. Han, J. Xu, Ever-reaching probabilities and mean first passage times of higher order ergodic Markov chains,Linear & Multilinear Algebra72: 59–75, 2024

2024