Searcharxiv⌕ Search

arXiv subjects

Projesh Nath Choudhury

Publications and source records attributed to Projesh Nath Choudhury.

At least 19 recordsLinked to original sources
arXiv2026

SNT-Rank: Kronecker Products and Euclidean Distance Matrices

Symmetric nonnegative matrix trifactorizations (SN-Trifactorizations) were introduced by Bukovšek-Šmigoc [Linear Algebra Appl. 2023] as a symmetric analogue of nonnegative matrix factorizations. A SN-Trifactorization of a symmetric nonnegative matrix $A$ is of the form $A = BCB^{T},$ where $B$ and $C$ are nonnegative matrices, with $C$ symmetric. The associated SNT-rank of $A$ is defined as the smallest integer $k$ for which $A$ admits such a factorization with $C \in \mathbb{R}_{+}^{k \times k}$. In this paper, we derive sharper upper bounds for the SNT-rank of the Euclidean distance matrices considered by Shitov [Linear Algebra Appl. 2025] and Bukovšek-Šmigoc [Linear Algebra Appl. 2023]. We also establish several new relationships between the rank and the SNT-rank of symmetric nonnegative matrices and show that the SNT-rank is submultiplicative with respect to the Kronecker product. Finally, motivated by a conjecture posed in the Dagstuhl Seminar Report 13082, we prove a multiplicativity result for the nonnegative rank under an additional structural assumption. We also partially resolve a conjecture of Vandaele-Gillis-Glineur-Tuyttens [J. Global Optim. 2016].

math.RA↗
arXiv2026

Quadratic Embedding Constants of Cartesian Products and Joins of Graphs

The quadratic embedding constant (QEC) of a finite, simple, connected graph originated from the classical work of Schoenberg [Ann. of Math., 1935] and [Trans. Amer. Math. Soc., 1938] on Euclidean distance geometry. In this article, we study the QEC of graphs in terms of two graph operations: the Cartesian product and the join of graphs. We derive a general formula for the QEC of the join of an arbitrary graph with a regular graph and with a complete multipartite graph. As an application of these results, we explicitly compute the QEC for several classes of graphs and provide new examples of graphs of QE class. We also establish a lower bound for the quadratic embedding constant of the Cartesian product of two arbitrary connected graphs. Furthermore, as an extremal case, we derive concise formulas for the quadratic embedding constants of the Cartesian product of an arbitrary graph G with a complete graph and with a complete bipartite graph, expressed in terms of $\qec(G)$.

math.CO↗
arXiv2026

Entrywise preservers of sign regularity

Entrywise functions preserving positivity and related notions have a rich history, beginning with the seminal works of Schur, Pólya-Szegő, Schoenberg, and Rudin. Following their classical results, it is well-known that entrywise functions preserving positive semidefiniteness for matrices of all dimensions must be real analytic with non-negative Taylor coefficients. These works were taken forward in the last decade by Belton, Guillot, Khare, Putinar, and Rajaratnam. Recently, Belton-Guillot-Khare-Putinar [J. d'Analyse Math. 2023] classified all functions that entrywise preserve totally positive (TP) and totally non-negative (TN) matrices. In this paper, we study entrywise preservers of strictly sign regular and sign regular matrices - a class that includes TP/TN matrices as special cases and was first studied by Schoenberg in 1930 to characterize variation diminution. Our main results provide complete characterizations of entrywise transforms of rectangular matrices which preserve: (i)~sign regularity and strict sign regularity, as well as (ii)~sign regularity and strict sign regularity with a given sign pattern.

math.FA↗
arXiv2026

Linear preserver problems in matrix positivity theory

Linear preserver problems have been a central focus of research in matrix theory and operator theory for more than a century, beginning with Frobenius' 1897 characterization of determinant preserving linear maps on the space of complex matrices. Since this foundational result, considerable work has examined linear preservers of diverse subsets, functions, and relations across different matrix and operator spaces. The purpose of this survey is to present the current state of research on linear preserver problems for several positivity classes of matrices. We provide an overview of recent developments in the literature and, for each positivity class considered, identify gaps that remain to guide future research.

math.FA↗
arXiv2026

Semigroup automorphisms of total positivity

Totally positive (TP) and totally nonnegative (TN) matrices connect to analysis, mechanics, and to dual canonical bases in reductive groups, by well-known works of Schoenberg, Gantmacher-Krein, Lusztig, and others. TP matrices form a multiplicatively closed semigroup, contained in the larger monoid of invertible totally nonnegative (ITN) matrices. Whitney and Berenstein-Fomin-Zelevinsky found bidiagonal factorizations of all $n\times n$ ITN and TP matrices into multiplicative generators; a natural question now is to classify the multiplicative automorphisms of these semigroups. In this article, we classify all automorphisms of these semigroups of ITN and TP matrices. In particular, we show that the automorphisms are the same, and they respect the multiplicative generators.

math.RA↗
arXiv2025

Positivity of GCD tensors and their determinants

Let $S=\{s_{1},s_{2},\ldots,s_{n}\}$ be an ordered set of $n$ distinct positive integers. The $m$th-order $n$-dimensional tensor $T_{[S]}=(t_{i_{1}i_{2}\ldots i_{m}}),$ where $t_{i_{1}i_{2}\ldots i_{m}}=GCD(s_{i_{1}},s_{i_{2}},\ldots,s_{i_{m}}),$ the greatest common divisor (GCD) of $s_{i_{1}},s_{i_{2}},\ldots,$ and $s_{i_{m}}$ is called the GCD tensor on $S$. The earliest result on GCD tensors goes back to Smith [Proc. Lond. Math. Soc., 1976], who computed the determinant of GCD matrix on $S=\{1,2,\ldots,n\}$ using the Euler's totient function, followed by Beslin-Ligh [Linear Algebra Appl., 1989] who showed all GCD matrices are positive definite. In this note, we study the positivity of higher-order tensors in the $k$-mode product. We show that all GCD tensors are strongly completely positive (CP). We then show that GCD tensors are infinite divisible. In fact, we prove that for every positive real number $r,$ the tensor $T_{[S]}^{\circ r}=(t^{r}_{i_{1}i_{2}\ldots i_{m}})$ is strongly CP. Finally, we obtain an interesting decomposition of GCD tensors using Euler's totient function $Φ$. Using this decomposition, we show that the determinant (also called hyperdeterminant) of the $m$th-order GCD tensor $T_{[S]}$ on a factor-closed set $S=\{s_1,\dots,s_n\}$ is $\prod\limits_{i=1}^{n} Φ(s_{i})^{(m-1)^{(n-1)}}$.

math.RA↗
arXiv2025

Sign regularity preserving linear operators

A matrix $A\in \mathbb{R}^{m \times n}$ is strictly sign regular/SSR (or sign regular/SR) if for each $1 \leq k \leq \min\{m,n\}$, all (non-zero) $k\times k$ minors of $A$ have the same sign. This class of matrices contains the totally positive matrices, and was first studied by Schoenberg in 1930 to characterize variation diminution, a fundamental property in total positivity theory. In this article, we classify all surjective linear mappings $\mathcal{L}:\mathbb{R}^{m\times n}\to\mathbb{R}^{m\times n}$ that preserve: (i) sign regularity and (ii) sign regularity with a given sign pattern, as well as (iii) strict versions of these.

math.FA↗
arXiv2025

Constructing strictly sign regular matrices of all sizes and sign patterns

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 $p \in [1, \min\{m,n\}]$.

math.RA↗
arXiv2024

Sign regular matrices and variation diminution: single-vector tests and characterizations, following Schoenberg, Gantmacher-Krein, and Motzkin

Variation diminution (VD) is a fundamental property in total positivity theory, first studied in 1912 by Fekete-Pólya for one-sided Pólya frequency sequences, followed by Schoenberg, and by Motzkin who characterized sign regular (SR) matrices using VD and some rank hypotheses. A classical theorem by Gantmacher-Krein characterized the strictly sign regular (SSR) $m \times n$ matrices for $m>n$ using this property. In this article we strengthen these results by characterizing all $m \times n$ SSR matrices using VD. We further characterize strict sign regularity of a given sign pattern in terms of VD together with a natural condition motivated by total positivity. We then refine Motzkin's characterization of SR matrices by omitting the rank condition and specifying the sign pattern. This concludes a line of investigation on VD started by Fekete-Pólya [Rend. Circ. Mat. Palermo 1912] and continued by Schoenberg [Math. Z. 1930], Motzkin [PhD thesis, 1936], Gantmacher-Krein [1950 book], Brown-Johnstone-MacGibbon [J. Amer. Stat. Assoc. 1981], and Choudhury [Bull. London Math. Soc. 2022, Bull. Sci. Math. 2023]. In fact we show stronger characterizations, by employing single test vectors with alternating sign coordinates - i.e., lying in the alternating bi-orthant. We also show that test vectors chosen from any other orthant will not work.

math.RA↗
arXiv2023

The blowup-polynomial of a metric space: connections to stable polynomials, graphs and their distance spectra

To every finite metric space $X$, including all connected unweighted graphs with the minimum edge-distance metric, we attach an invariant that we call its blowup-polynomial $p_X(\{ n_x : x \in X \})$. This is obtained from the blowup $X[{\bf n}]$ - which contains $n_x$ copies of each point $x$ - by computing the determinant of the distance matrix of $X[{\bf n}]$ and removing an exponential factor. We prove that as a function of the sizes $n_x$, $p_X({\bf n})$ is a polynomial, is multi-affine, and is real-stable. This naturally associates a hitherto unstudied delta-matroid to each metric space $X$; we produce another novel delta-matroid for each tree, which interestingly does not generalize to all graphs. We next specialize to the case of $X = G$ a connected unweighted graph - so $p_G$ is "partially symmetric" in $\{ n_v : v \in V(G) \}$ - and show three further results: (a) We show that the polynomial $p_G$ is indeed a graph invariant, in that $p_G$ and its symmetries recover the graph $G$ and its isometries, respectively. (b) We show that the univariate specialization $u_G(x) := p_G(x,\dots,x)$ is a transform of the characteristic polynomial of the distance matrix $D_G$; this connects the blowup-polynomial of $G$ to the well-studied "distance spectrum" of $G$. (c) We obtain a novel characterization of complete multipartite graphs, as precisely those for which the "homogenization at $-1$" of $p_G({\bf n})$ is real-stable (equivalently, Lorentzian, or strongly/completely log-concave), if and only if the normalization of $p_G(-{\bf n})$ is strongly Rayleigh.

math.MG↗
arXiv2023

The additive-multiplicative distance matrix of a graph, and a novel third invariant

Graham showed with Pollak and Hoffman-Hosoya that for any directed graph $G$ with strong blocks $G_e$, the determinant $\det(D_G)$ and cofactor-sum $cof(D_G)$ of the distance matrix $D_G$ can be computed from the same quantities for the blocks $G_e$. This was extended to trees - and in our recent work to any graph - with multiplicative and $q$-distance matrices. For trees, we went further and unified all previous variants with weights in a unital commutative ring, into a distance matrix with additive and multiplicative edge-data. In this work: (1) We introduce the additive-multiplicative distance matrix $D_G$ of every strongly connected graph $G$, using what we term the additive-multiplicative block-datum $\mathcal{G}$. This subsumes the previously studied additive, multiplicative, and $q$-distances for all graphs. (2) We introduce an invariant $κ(D_G)$ that seems novel to date, and use it to show "master" Graham-Hoffman-Hosoya (GHH) identities, which express $\det(D_G), cof(D_G)$ in terms of the blocks $G_e$. We show how these imply all previous variants. (3) We show $\det(.), cof(.), κ(.)$ depend only on the block-data for not just $D_G$, but also several minors of $D_G$. This was not studied in any setting to date; we show it in the "most general" additive-multiplicative setting, hence in all known settings. (4) We compute $D_G^{-1}$ in closed-form; this specializes to all known variants. In particular, we recover our previous formula for $D_T^{-1}$ for additive-multiplicative trees (which itself specializes to a result of Graham-Lovasz and answers a 2006 question of Bapat-Lal-Pati.) (5) We also show that not the Laplacian, but a closely related matrix is the "correct" one to use in $D_G^{-1}$ - for the most general additive-multiplicative matrix $D_G$ of each $G$. As examples, we compute in closed form $\det(D_G), cof(D_G), κ(D_G), D_G^{-1}$ for hypertrees.

math.CO↗
arXiv2023

Distance matrices of a tree: two more invariants, and in a unified framework

Graham-Pollak showed that for $D = D_T$ the distance matrix of a tree $T$, det$(D)$ depends only on its number of edges. Several other variants of $D$, including directed/multiplicative/$q$- versions were studied, and always, det$(D)$ depends only on the edge-data. We introduce a general framework for bi-directed weighted trees, with threefold significance. First, we improve on state-of-the-art for all known variants, even in the classical Graham-Pollak case: we delete arbitrary pendant nodes (and more general subsets) from the rows/columns of $D$, and show these minors do not depend on the tree-structure. Second, our setting unifies all known variants (with entries in a commutative ring). We further compute $D^{-1}$ in closed form, extending a result of Graham-Lovasz [Adv. Math. 1978] and answering an open question of Bapat-Lal-Pati [Lin. Alg. Appl. 2006] in greater generality. Third, we compute a second function of the matrix $D$: the sum of all its cofactors, cof$(D)$. This was worked out in the simplest setting by Graham-Hoffman-Hosoya (1978), but is relatively unexplored for other variants. We prove a stronger result, in our general setting, by computing cof$(.)$ for minors as above, and showing these too depend only on the edge-data. Finally, we show our setting is the "most general possible", in that with more freedom in the edgeweights, det$(D)$ and cof$(D)$ depend on the tree structure. In a sense, this completes the study of the invariants det$(D_T)$, cof$(D_T)$ for trees $T$ with edge-data in a commutative ring. Moreover: for a bi-directed graph $G$ we prove multiplicative Graham-Hoffman-Hosoya type formulas for det$(D_G)$, cof$(D_G)$, $D_G^{-1}$. We then show how this subsumes their 1978 result. The final section introduces and computes a third, novel invariant for trees and a Graham-Hoffman-Hosoya type result for our "most general" distance matrix $D_T$.

math.CO↗
arXiv2023

Quadratic embedding constants of graphs: Bounds and distance spectra

The quadratic embedding constant (QEC) of a finite, simple, connected graph $G$ is the maximum of the quadratic form of the distance matrix of $G$ on the subset of the unit sphere orthogonal to the all-ones vector. The study of these QECs was motivated by the classical work of Schoenberg on quadratic embedding of metric spaces [Ann. of Math., 1935] and [Trans. Amer. Math. Soc., 1938]. In this article, we provide sharp upper and lower bounds for the QEC of trees. We next explore the relation between distance spectra and quadratic embedding constants of graphs - and show two further results: $(i)$ We show that the quadratic embedding constant of a graph is zero if and only if its second largest distance eigenvalue is zero. $(ii)$ We identify a new subclass of nonsingular graphs whose QEC is the second largest distance eigenvalue. Finally, we show that the QEC of the cluster of an arbitrary graph $G$ with either a complete or star graph can be computed in terms of the QEC of $G$. As an application of this result, we provide new families of examples of graphs of QE class.

math.CO↗
arXiv2023

Total negativity: Characterizations and single-vector tests

A matrix is called totally negative (totally non-positive) of order $k$, if all its minors of size at most $k$ are negative (non-positive). The objective of this article is to provide several novel characterizations of total negativity via the (a) sign non-reversal property, (b) variation diminishing property, and (c) Linear Complementarity Problem. More strongly, each of these three characterizations uses a single test vector. As an application of the sign non-reversal property, we study the interval hull of two rectangular matrices. In particular, we identify two matrices $C^\pm(A,B)$ in the interval hull of matrices $A$ and $B$ that test total negativity of order $k$, simultaneously for the entire interval hull. We also show analogous characterizations for totally non-positive matrices. These novel characterizations may be considered similar in spirit to fundamental results characterizing totally positive matrices by Brown--Johnstone--MacGibbon [J. Amer. Statist. Assoc. 1981] (see also Gantmacher--Krein, 1950), Choudhury--Kannan--Khare [Bull. London Math. Soc., 2021] and Choudhury [Bull. London Math. Soc., 2022]. Finally using a 1950 result of Gantmacher--Krein, we show that totally negative/non-positive matrices can not be detected by (single) test vectors from orthants other than the open bi-orthant that have coordinates with alternating signs, via the sign non-reversal property or the variation diminishing property.

math.RA↗
arXiv2022

Blowup polynomials and delta-matroids of graphs

For every finite simple connected graph $G = (V,E)$, we introduce an invariant, its blowup-polynomial $p_G(\{ n_v : v \in V \})$. This is obtained by dividing the determinant of the distance matrix of its blowup graph $G[{\bf n}]$ (containing $n_v$ copies of $v$) by an exponential factor. We show that $p_G({\bf n})$ is indeed a polynomial function in the sizes $n_v$, which is moreover multi-affine and real-stable. This associates a hitherto unexplored delta-matroid to each graph $G$; and we provide a second unexplored one for each tree. As another consequence, we obtain a new characterization of complete multipartite graphs, via the homogenization at $-1$ of $p_G$ being completely/strongly log-concave, i.e., Lorentzian. (These results extend to weighted graphs.) Finally, we show $p_G$ is indeed a graph invariant, i.e., $p_G$ and its symmetries (in the variables ${\bf n}$) recover $G$ and its isometries, respectively.

math.CO↗
arXiv2021

Characterizing total positivity: single vector tests via Linear Complementarity, sign non-reversal, and variation diminution

A matrix $A$ is called totally positive (or totally non-negative) of order $k$, denoted by TP_k (or TN_k), if all minors of size at most $k$ are positive (or non-negative). These matrices have featured in diverse areas in mathematics, including algebra, analysis, combinatorics, and probability theory. The goal of this article is to provide a novel connection between total positivity and optimization/game theory. Specifically, we draw a relationship between TP matrices and the Linear Complementarity Problem (LCP), which generalizes and unifies linear and quadratic programming problems and bimatrix games - this connection is unexplored, to the best of our knowledge. We show that $A$ is $TP_k$ if and only if for every contiguous square submatrix $A_r$ of $A$, $LCP(A_r,q)$ has a unique solution for each vector $q<0$. In fact this can be strengthened to check the solution set of LCP at a single vector for each such square submatrix. These novel characterizations are in the spirit of classical results characterizing $TP$ matrices by Gantmacher-Krein [Compos. Math. 1937] and P-matrices by Ingleton [Proc. London Math. Soc. 1966]. Our work contains two other contributions, both of which characterize TP using single test vectors. First, we improve on one of the main results in recent joint work [Bull. London Math. Soc., 2021], which provided a novel characterization of TP_k matrices using sign non-reversal phenomena. We further improve on a classical characterization of TP by Brown-Johnstone-MacGibbon [J. Amer. Statist. Assoc. 1981] (following Gantmacher-Krein, 1950) involving the variation diminishing property. Finally, we use a Pólya frequency function of Karlin [Trans. Amer. Math. Soc. 1964] to show that our aforementioned characterizations of TP, involving test-vectors drawn from the `alternating' bi-orthant, do not work if these vectors are drawn from any other open orthant.

math.RA↗
arXiv2021

Multivariate blowup-polynomials of graphs

In recent joint work (2021), we introduced a novel multivariate polynomial attached to every metric space - in particular, to every finite simple connected graph $G$ - and showed it has several attractive properties. First, it is multi-affine and real-stable (leading to a hitherto unstudied delta-matroid for each graph $G$). Second, the polynomial specializes to (a transform of) the characteristic polynomial $χ_{D_G}$ of the distance matrix $D_G$; as well as recovers the entire graph, where $χ_{D_G}$ cannot do so. Third, the polynomial encodes the determinants of a family of graphs formed from $G$, called the blowups of $G$. In this short note, we exhibit the applicability of these tools and techniques to other graph-matrices and their characteristic polynomials. As a particular case, we will see that the adjacency characteristic polynomial $χ_{A_G}$ is in fact the shadow of a richer multivariate blowup-polynomial, which is similarly multi-affine and real-stable. Moreover, this polynomial encodes not only the aforementioned three properties, but also yields additional information for specific families of graphs.

math.CO↗