Constructing strictly sign regular matrices of all sizes and sign patterns

Projesh Nath Choudhury; Shivangi Yadav

REVIEW 2 major objections 5 minor 17 references

Constructing strictly sign regular matrices of all sizes and sign patterns

T0 review · 2 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read This paper proves that every strictly sign regular matrix can be extended by a new row or column on any border, and uses that fact to build an algorithm that constructs a strictly sign regular matrix of any specified size and sign pattern.

desk verdict A genuinely useful explicit construction for SSR matrices; the main theorem has a repairable gap in the rectangular border case and the Python code has a 1D orientation bug, but the core idea is sound and worth refereeing. read the letter →

arxiv 2411.14287 v2 pith:CTUSP7FQ submitted 2024-11-21 math.RA

Projesh Nath Choudhury , Shivangi Yadav This is my paper · ORCID

classification math.RA MSC 15B4815A8315A1515-04

keywords strictlysignregularmatricestotallypositivepatternmatrixcompletionconstructionalgorithmcontiguousminorslineinsertion

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

A strictly sign regular (SSR) matrix is one whose subdeterminants of any fixed size are either all positive or all negative. The paper shows that any SSR matrix can be extended by adding one row or column to any of its four borders, and that a new line can also be inserted between any two existing lines, keeping the matrix SSR. It then turns this extension step into an algorithm that, given any desired dimensions and sign pattern, outputs an explicit SSR matrix, with Python code supplied. The constructive step matters because classical results guarantee such matrices exist but give no way to write one down; explicit examples are needed across applications that use total positivity and its sign-regular cousins.

What carries the argument

The load-bearing object is the normalized new-column representation $c=\sum_{i=1}^{r}(-1)^{i-1}y_i c_i$ with $y_i>0$, where $r=\min(m,n)$. A lemma shows every column of an SSR matrix has such an alternating form, so the search for a valid extension reduces to choosing the positive coefficients $y_i$; the proof sets $y_n$ arbitrarily and then chooses $y_k$ one by one via finitely many explicit lower bounds (equation (2.4)) so that all $k\times k$ contiguous minors containing the new line keep the sign $\epsilon_k$. A standard theorem (quoted in the paper) reduces SSR to checking contiguous minors, and the exchange matrix $P_n$ transposes the construction from left border to right border and from top to bottom. For the tall case $m>n$, the additional ingredient is a perturbation of the first $m-n$ entries of the new column by amounts smaller than $\lambda_k/\Lambda_k$, the ratio of the smallest to the largest modulus of the relevant nonzero contiguous minors, which preserves existing signs while forcing the new larger minors to be nonzero and equisigned.

What would settle it

Run the supplied Python implementation to produce a $4\times 5$ SSR matrix with sign pattern $(1,1,-1,-1)$, then verify in exact arithmetic that every contiguous minor of size $1,2,3,4$ carries the claimed sign; any sign violation would refute the construction. A sharper test: take a known $3\times 3$ SSR matrix, add a column on the right with Algorithm 1, and check that all $3\times 3$ contiguous minors of the $3\times 4$ result have the sign required by the given pattern.

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Extended reading notes

Core claim

The central discovery is that border extension is always possible: for an $m\times n$ SSR matrix with sign pattern $\epsilon$, there is a new column (or row) whose addition on any chosen side keeps the matrix SSR with the same signs for minors of existing sizes, and the sign of any newly appearing larger minors can be chosen freely. The new column is constructed in normalized form $c=\sum_{i=1}^{\min(m,n)}(-1)^{i-1}y_i c_i$ with all $y_i>0$, choosing $y_n,y_{n-1},\ldots$ one at a time so that every contiguous minor containing the new line has the required sign; only finitely many lower bounds are needed at each step. In the rectangular case $m>n$, the same coefficients make the matrix SSR of order $n$, and a controlled perturbation of the first $m-n$ entries of the new column, with step sizes below the ratio of the smallest to the largest relevant minor modulus, makes all $(n+1)$-minors nonzero with the prescribed sign. The same machinery inserts a line between consecutive lines (by completing the matrix to a square even-order SSR matrix, inserting in the middle, and deleting the added lines), and the whole procedure yields Algorithms 3 and 4 for any size, sign pattern, and order $p$.

Load-bearing premise

The construction relies on the premise that the new line can always be written as an alternating sum of existing lines with positive coefficients, and that in tall matrices one can nudge the first entries of that line by tiny amounts—no larger than the ratio of the smallest to the largest relevant subdeterminant—without changing the sign of any existing minor.

Editorial extensions

If this is right

Every SSR matrix can be grown, one line at a time, into an SSR matrix of any larger size with the same sign pattern on the old minors and freely chosen signs on new sizes.
A line can be inserted between any two consecutive rows or columns, so the construction can also refine an SSR matrix without disturbing its order of lines.
For any $p\le\min(m,n)$ and any sign pattern of length $p$, an $m\times n$ SSR$_p$ matrix exists and is produced by the algorithm.
The same algorithm covers the totally positive case (all signs $+1$), giving explicit totally positive matrices of arbitrary size, with the insertion step matching an earlier construction for that special case.
Rectangular matrices taller than wide can still be extended, with the new larger minors forced to be either all positive or all negative at will.

Reading between the lines

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

The recursive lower-bound step could be adapted to generate random SSR matrices by sampling the positive coefficients $y_i$; such a sampler would give statistically usable test sets for numerical experiments that currently rely on a handful of known examples.
Because Theorem A lets one extend on all four borders, the SSR matrices of fixed size form a connected graph under single-line extensions, which could support constructive proofs or searches that need to move continuously through the class.
The perturbation argument for the tall case suggests a quantitative stability statement: every SSR matrix of order $n$ that sits inside a taller matrix can be completed to an SSR matrix of order $n+1$ as long as the gap between the smallest and largest relevant minors is not too small; this might transfer to nearby matrices with approximate signs.
The same framework may apply to other equisigned minor classes, such as sign-regular (non-strict) matrices, by taking limits of the perturbations, although the paper does not pursue that direction.

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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

Two things to know. First, this paper delivers a genuinely new explicit construction: SSR matrices of any size and any sign pattern, built by border extension and interior line insertion, with an algorithm and Python code. The earlier literature had existence arguments (Gantmacher–Krein) and a totally positive-only line insertion result (Johnson–Smith), so the general sign-pattern version is a real step forward. Second, the main construction probably works, but the proof of Theorem 2.7 is not fully rigorous as written. The odd-l perturbation case is left to “a similar argument,” and that is exactly the case where the δ-term opposes the existing minor sign. The missing bound is easy to supply with the same λ_k, Λ_k argument used for even l, so the paper is repairable, but the gap is load-bearing for rectangular border extension.

What the paper does well: the square case (Theorem 2.5) is proved cleanly by inductively choosing positive coefficients from finite lower bounds. The interior line insertion theorem (Theorem 3.1) is a genuine extension beyond the TP case, and the reduction in Theorem B is elegant. The algorithm section is clear, and shipping actual Python code is a plus, not a formality.

Soft spots, in proportion to how soft they are. The Theorem 2.7 omission is real but fixable; I would not reject over it. Equation (2.3) has a harmless sign typo in the displayed formula versus the lower bound that follows it. The Appendix A code has a concrete bug in the min_dim=1 branch: it builds a 1×max_dim row and then returns the transpose when m=1, so the output dimensions are swapped for one-dimensional matrices. The SSR_p results are sketched rather than fully detailed; that is acceptable for a companion result but should be checked line by line. There are also assorted typos and OCR-style artifacts in the text.

Citation pattern is fine. The self-citations in the introduction motivate the question but are not load-bearing premises, and the proofs rely on Karlin’s theorem, Cramer’s rule, and determinant identities rather than on the authors’ own earlier results.

Who is this for? Anyone working with SSR matrices in analysis, splines, combinatorics, or Gabor frames who needs concrete examples. It deserves a serious referee: the central construction is novel and plausible, and the gaps are repairable. My recommendation is to send it to review with a request for a revised version that fills the odd-l cancellation bound in Theorem 2.7, fixes the sign typo, and corrects the Python code’s 1D branch.

Referee Report

2 major / 5 minor

Summary. The paper proposes an explicit construction of strictly sign regular matrices of arbitrary size and sign pattern. The main tool is a border-extension theorem (Theorem A): every SSR matrix can be bordered by a new row or column while remaining SSR, with control over the sign of any newly appearing larger minors. A second theorem (Theorem B) inserts a line between two consecutive lines of an SSR matrix. The proofs in Section 2 choose the new line as an alternating linear combination of existing lines with sufficiently large positive coefficients, and in the rectangular case perturb the first entries of the new column to make the larger minors nonzero with the desired sign. Section 3 handles middle insertions by completing the matrix to an even square, and Section 4 converts the constructions into Algorithms 1-4 with Python code in Appendix A.

Significance. If the identified gaps are repaired, the paper gives the first fully constructive algorithm for SSR matrices of every size and sign pattern, answering a natural question left open by the classical existence results of Gantmacher-Krein. The reduction to Karlin's contiguous-minor criterion is appropriate, and the coefficient choices are finite and computable. The extension and insertion theorems are also of independent interest for generating test matrices in total positivity and variation-diminishing applications. The proofs use Karlin's theorem, Cramer's rule, and determinant identities, and no circular dependence on the paper's own earlier results was found. However, the computational appendix is not yet a reliable implementation of the stated algorithms, and one case in the proof of Theorem 2.7 is left implicit.

major comments (2)

[Section 2, proof of Theorem 2.7] The proof of sign preservation for r x r minors containing the kth row and first column is incomplete for odd l. In the displayed expansion, when l is odd the delta_k term is added with coefficient det(Â^{δ_{k−1}})_{I\{i_l}×[r]\{1}}; after the identity for that minor, the delta_k contribution has sign ε_{r−1}. If ε_{r−1}=ε_r there is nothing to prove, but if ε_{r−1}≠ε_r the two terms are opposite and one must use the bound ε_r det(Â^{δ_k})_{I×[r]} ≥ λ_k − δ_kΛ_k > 0. The text writes this bound only for even l and says the odd case is similar. Since this inequality is exactly what keeps the construction inside SSR_n(ε), it should be stated explicitly. The missing argument is short and uses the already-defined λ_k and Λ_k, so the gap is repairable.
[Appendix A, Listings 4-5] The Python code does not currently implement Algorithms 2-4 as described. In SSR_construction, the min_dim=1 branch tests `min_dim == n` and otherwise transposes, returning the wrong orientation (for example, m=1,n=3 becomes 3 x 1 and m=3,n=1 becomes 1 x 3); the test should be `min_dim == m`. In SSR_p_construction, the first loop transposes A after every call to add_col_left, so starting from p x p it does not produce a p x m matrix; for instance p=2,m=5,n=5 yields a 4 x 6 matrix rather than 5 x 5. In add_perturbation, the update of λ and Λ with the matrix B is executed only for k>1, so the modulus 1 of the 1 x 1 minor of B is omitted from Λ, and the chosen perturbation can be larger than the intended λ/Λ bound. These deviations affect the advertised reproducibility of the algorithm and should be corrected.

minor comments (5)

[Section 2, Theorem A proof] The one-dimensional cases (n=1 or m=1) are not covered by Lemma 2.2 or by the rectangular perturbation argument; they should be dispatched explicitly, for example by taking a constant vector of the required sign.
[Section 2, Theorem A proof] When m=n, the top- and bottom-row insertions are described via Theorem 2.7 applied to A^T, but Theorem 2.7 assumes rows > columns; Theorem 2.5 or Remark 2.6 should be cited for the square case.
[Theorem 2.7, r=1 case] The phrase 'if ε_{r−1}=ε_r' needs a convention for ε_0, or a separate sentence for the 1 x 1 minors, since the proof uses this comparison already for r=1.
[Throughout Section 2] The notation '[r]/integerdivide{1}' appears to be an OCR or typesetting corruption of set difference; the displayed determinants should be typeset as [r]\{1} and [n+1]\{1}.
[Theorem 3.2] For p=1, the proposed column formula c = Σ_{i=1}^{⌈p/2⌉}(−1)^{i−1}y_i(c_{p−i}+c_{p+i−1}) refers to c_0 and is undefined; the p=1 case should be treated separately by taking any vector with the required sign.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular derivation found: the border-extension and line-insertion proofs rely on Karlin's contiguous-minor theorem, Cramer's rule, and determinant identities; self-citations are motivational only.

full rationale

The central derivation chain is self-contained and does not reduce to its inputs. Theorem A is proved in Section 2 by constructing the new bordering line as an alternating linear combination of existing columns (Lemma 2.2), then choosing the coefficients sequentially from finite lower bounds derived by determinant expansions (equations (2.3) and (2.4)); the only external structural input is Karlin's theorem that it suffices to check contiguous minors, which is an independent classical result. The rectangular case in Theorem 2.7 uses an explicit perturbation argument with moduli bounded by lambda_k/Lambda_k, and the signs of the (n+1)x(n+1) minors are controlled directly. Theorem B is derived from Theorem A and Theorem 3.1, where the inserted line is again produced as a linear combination of existing columns with inductively chosen positive coefficients. The algorithm in Section 4 is a direct transcription of these constructions, not a fit of the target sign pattern into the construction. The only self-citations, [6] and [7], appear in the introduction to motivate the problem and are not used as premises in any proof; the proofs instead invoke Karlin [13] and elementary linear algebra. The possible gap in Theorem 2.7 for odd l (the implicit 'similar argument') is an incompleteness or exposition issue about a repairable inequality, not a circularity: the needed bound is of the same type as the even-l case and does not assume the conclusion. No fitted quantity is renamed as a prediction, no uniqueness theorem is imported from the authors' own work, and no known result is merely relabeled. Therefore the circularity score is 0.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

No constants are fitted to data and no new physical or mathematical entities are postulated. The y_i and delta_i appearing in the construction are existence variables chosen from finite lower bounds, not free parameters calibrated to external measurements. The main external input is Karlin's standard contiguous-minor theorem.

assumptions (3)

standard math Karlin's contiguous-minor criterion: an m x n matrix is SSRp with sign pattern eps iff all k x k contiguous minors have sign eps_k for every k <= p.
Invoked as Theorem 2.4 and used throughout to reduce sign regularity checking to contiguous minors.
standard math Determinant multilinearity, Cramer's rule, and Cauchy-Binet identities.
Used in Lemma 2.2 and in the coefficient lower-bound derivations in Sections 2 and 3.
standard math Submatrices of SSR matrices are SSR.
Used in Theorem B after removing the extra rows and columns added during the square completion.

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Pith. "Pith review of Constructing strictly sign regular matrices of all sizes and sign patterns." pith.science (2026). https://pith.science/paper/CTUSP7FQ

@misc{pith2026241114287,
  author       = {Pith},
  title        = {Pith review of: Constructing strictly sign regular matrices of all sizes and sign patterns},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CTUSP7FQ}},
  note         = {Machine review of arXiv:2411.14287}
}

abstract

The class of strictly sign regular (SSR) matrices has been extensively studied by many authors over the past century, notably by Schoenberg, Motzkin, Gantmacher, and Krein. A classical result of Gantmacher-Krein assures the existence of SSR matrices for any dimension and sign pattern. In this article, we provide an algorithm to explicitly construct an SSR matrix of any given size and sign pattern. (We also provide in an Appendix, a Python code implementing our algorithm.) To develop this algorithm, we show that one can extend an SSR matrix by adding an extra row (column) to its border, resulting in a higher order SSR matrix. Furthermore, we show how inserting a suitable new row/column between any two successive rows/columns of an SSR matrix results in a matrix that remains SSR. We also establish analogous results for strictly sign regular $m \times n$ matrices of order $p$ for any $p \in [1, \min\{m,n\}]$.

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Reference graph

Works this paper leans on

17 extracted references · 16 canonical work pages

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M.J. Tsatsomeros and Y.F. Zhang. The ﬁbre of P-matrices: the recursive construction of all ma trices with positive principal minors . Linear Multilinear Algebra , 69:224–232, 2021. Appendix A. Python code Here, we include a Python program for constructing an m × n SSR matrix o...

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Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [1]

Abascal, F

P. Abascal, F. Fueyo, J. Jimenez, A. Palacio, and M.L. Ser rano. Characterization and construction of sign regular tridiagonal matrices. Linear Multilinear Algebra , published online, 2024

work page 2024

[2] [2]

Belton, D

A. Belton, D. Guillot, A. Khare, and M. Putinar. Totally p ositive kernels, P´ olya frequency functions, and their transforms. J. d’Analyse Math. , 150:83–158, 2023

work page 2023

[3] [3]

Berenstein, S

A. Berenstein, S. Fomin, and A. Zelevinsky. Parametriza tions of canonical bases and totally positive matrices. Adv. Math. , 122:49–149, 1996

work page 1996

[4] [4]

F. Brenti. Combinatorics and total positivity. J. Comb. Theory Ser. A, 71(2):175–218, 1995

work page 1995

[5] [5]

Brown, I.M

L.D. Brown, I.M. Johnstone, and K.B. MacGibbon. Variati on diminishing transformations: a direct approach to total positivity and its statistical applications. J. Amer. Statist. Assoc. , 76(376):824–832, 1981

work page 1981

[6] [6]

Choudhury and S

P.N. Choudhury and S. Yadav. Sign regular matrices and va riation diminution: single-vector tests and char- acterizations, following Schoenberg, Gantmacher–Krein, and Motzkin. Proc. Amer. Math. Soc. , 153:497–511, 2025

work page 2025

[7] [7]

Choudhury and S

P.N. Choudhury and S. Yadav. Sign regularity preserving linear operators. Preprint, arXiv:2408.02428, 2024

arXiv 2024

[8] [8]

Fekete and G

M. Fekete and G. P´ olya. ¨Uber ein Problem von Laguerre. Rend. Circ. Mat. Palermo , 34:89–120, 1912

work page 1912