Two decompositions of triangular matrices over rings

Abstract In this paper, we study triangular matrix categories by using the theory of recollements of abelian categories. Given a triangular matrix category, we construct two canonical recollements. We show that if certain functors of these recollements are exact, then the category appearing in the middle term is actually a category of modules over a triangular matrix category. This result is a generalization of one given by Li (Commun Algebra 46(2):615–628, 2017. https://doi.org/10.1080/00927872.2017.1327051 ). Finally, we show that if $$\textrm{Mod}(\mathcal {C})$$ Mod ( C ) admits a nontrivial torsion pair by abelian categories, then $$\mathcal {C}$$ C is equivalent to a triangular matrix category.

The viability and market value of pumpkin seeds are critically dependent on their internal plumpness, which is a comprehensive indicator of seed quality and can be compromised by factors such as hollow kernels resulting from improper storage or processing. Traditional methods for assessing internal quality are often destructive, time-consuming, and inefficient. In this study, the internal quality of pumpkin seeds is evaluated for the first time using the nondestructive terahertz (THz) time-domain imaging system combining compressed sensing (CS) with the Real-World Enhanced Super-Resolution Generative Adversarial Network (Real-ESRGAN) method. Using the Alternating Direction Method of Multipliers-Total Variation (ADMM_TV) as the reconstruction algorithm, five measurement matrices, including the Gauss Matrix (GaussMtx) and the Part-Fourier Matrix (PartFourierMtx), were compared. After selecting GaussMtx, the performance of five reconstruction algorithms, including Basis Pursuit (BP) and Stagewise Weak Orthogonal Matching Pursuit (SWOMP), was further compared. The super-resolution reconstruction of THz images reconstructed by CS was performed through Real-ESRGAN. The image quality was evaluated by objective indicators, and the detection accuracy was verified by the plumpness error. The results indicate that the combination of GaussMtx and ADMM_TV performs best in terms of PSNR, NMSE, and SSIM. After processing with Real-ESRGAN, the edge sharpness and details of THz images were significantly improved. The fullness error was only 2.23%, and the average detection error on the verification set was 3.37%. In summary, the combination of GaussMtx and ADMM_TV reconstruction algorithm, followed by super-resolution processing with Real-ESRGAN, can effectively improve the efficiency and accuracy of pumpkin seed quality detection. PRACTICAL APPLICATIONS: This research enables seed companies and food processors to quickly and accurately identify plump, high-quality pumpkin seeds without damaging them. By using an advanced imaging technique, the method can help automate quality control on production lines, ensuring better seed selection for planting and more consistent product quality for consumers. This contributes to reducing waste and improving the overall value of the agricultural products.

Nuclear data consist of nuclide-specific files that tabulate a wide range of quantities, including reaction cross sections, scattering and absorption probabilities, and decay coefficients. Here we reinterpret this body of data through a graph-theoretic lens, representing it algebraically as an adjacency matrix. We then analyze how the structural and spectral properties of this matrix influence numerical solutions of the nuclear inventory equation, with particular emphasis on the Chebyshev rational approximation method and the backward differentiation formula methods. Activation-decay simulations of nuclear inventories typically proceed in two stages: an activation phase under irradiation, followed by a decay phase after the source is removed. While both phases solve the same stiff system of linear ordinary differential equations, the decay phase has a distinctive property: the transmutation graph becomes acyclic. We exploit this by applying a topological ordering of isotopes, which transforms the decay matrix into strictly triangular form. This removes the need for expensive LU factorizations during the decay phase, replacing it with a single forward or back-substitution step. The result is a gain in computational speed with increased accuracy.

In this paper we investigate the numerical range of 3×3 matrices over finite fields, particularly when the matrix is strictly triangular. We provide a conjecture for this case that extends to n×n matrices for n ≥ 3 and also provide sample code for generating the numerical range.

Constructing an enhanced variant of Hill cipher based on 2D hyper chaotic map over GF(2^ <i>n</i> )

In this paper, we solve the problem of describing the coordinate transformations that preserve an upper triangular Toeplitz form of a given operator field. This problem is of fundamental importance in geometry, and its solution yields auxiliary transformations for the corresponding nondiagonalizable quasilinear systems. Surprisingly, this problem is closely related to the description of all Nijenhuis operators in the same form. This description, as well as the formulas for the aforementioned coordinate transformations, are given by the implicit formulas involving matrix-valued functions.

Regular triangular forms of rank exceeding 3

The Cayley transform of a square matrix $A$, defined as $F=(I+A)^{-1}(I-A)$, is tantamount to the factorization $A=(I+F)^{-1}(I-F)$. In this context, Fallat {\it et al.} (Electron. J. Linear Algebra, 9:190-196, 2002) and Mondal {\it et al.} (Linear Algebra Appl., 681:1-20, 2024) studied the Cayley transform of matrix positivity classes, namely, $P$-matrices, positive definite matrices, as well as $H$-matrices, $M$-matrices, and their inverse classes. The Cayley transform of $J$-symplectic, Toeplitz and dual matrices has also been considered in the literature. In this paper, the discussion is extended by examining the Cayley transform of positive semidefinite matrices, $Q^*$-matrices, EP-matrices, weighted-EP matrices, GP matrices, idempotent matrices, $T$-Hermitian matrices, $T$-EP matrices, $S$-skew symmetric matrices, $S$-normal matrices, centrosymmetric matrices, tridiagonal matrices, block triangular matrices, and semiconvergent matrices. The results complement the existing literature and are illustrated with examples. Connections among the matrix classes considered are also discussed and a summary of all results on the matrix Cayley transform known to date is compiled in the form of a table.

The fuzzy matrices play an important role in various fields.In this research paper we discussed some elementary operations on proposed triangular fuzzy numbers (TFNs).We also defined some fuzzy arithmetic operations on triangular fuzzy matrices (TFMs).A description on trace of triangular fuzzy matrices (TTFMs) is proposed.Some more special properties of trace of triangular fuzzy matrices have also been discussed.Some applications of Numerical example are also provided here.

We study commutative local rings over which every upper-triangular matrix is the sum of an idempotent and a $q$-potent that commute. For Galois rings and rings of the form $\mathbb{F}_{p^{k}}[x]/\langle x^{r} \rangle$, necessary and sufficient criterion are provided.

ABSTRACT This paper presents a linearly polarized, ground‐defected 2 × 2 rectangular patch antenna array with microstrip inset feeding. A simple structure is realized by placing the rectangular patches and the feeding network on the same layer. The proposed microstrip inset‐fed configuration facilitates proper impedance matching and easy array formation. This work provides an in‐depth examination of the antenna array's characteristics and performance. The antenna array is fabricated on an RT Duroid substrate and integrates a dumbbell‐shaped defected ground structure (DGS) in a triangular form with four radiating elements. The primary objective of incorporating the DGS is to improve the return loss. The proposed compact antenna array has overall dimensions of 106.3 × 112 × 1.575 mm 3 and is intended for wireless communication applications. The designed array operates over the desired frequency band of 3.52–3.61 GHz, achieving a reflection coefficient better than −10 dB and an impedance bandwidth of 25.09%. The antenna exhibits stable radiation characteristics with a peak gain of approximately 13.08 dBi across the operating band. The array is analyzed using HFSS, and fabrication and measurements are carried out to validate the simulated results. The measured outcomes show good agreement with the simulations, demonstrating the suitability of the proposed antenna for 5G midband mobile broadband wireless communication applications.

This paper introduces the Gaussian Lehmer sequence and presents its representation using a lower triangular Pascal matrix. Building on this matrix form, a new coding method is developed with improved capabilities for error detection and correction. The study explores the use of this coding approach in the realm of complex numbers within coding theory. Further investigation shows that the sequence follows distinct recurrence relations for odd and even terms. To unify these behaviors, a new function is proposed that captures both cases, resulting in the definition of a biperiodic sequence. The paper also provides a thorough analysis of the circulant matrix generated by the sequence, including the computation of its determinant and inverse, highlighting the promise of this matrix-based framework in advancing coding theory.

The Harary Index is an important topological parameter for examining the structure of a graph. This work presents a quantitative analysis of the structural features of zero-divisor graph using the Harary Index. In this research article, we compute the Harary Index for the zero-divisor graph of the ring of upper triangular matrices over a finite commutative ring and non commutative ring. In this study, we have computed the Harary index for directed and undirected zero-divisor graph obtained from upper triangular matrix [Formula: see text].

IOur focus is on the set of lower-triangular, infinite matrices that have natural operations like addition, multiplication by a number, and matrix multiplication. With respect to addition this set forms and abelian group while with respect to matrix multiplication, the invertivle elements of the set form a group. The set becomes an algebra (non-commutative in fact) with unity when all three operations are considered together. We indicate important properties of the algebraic structures obtained in this way. In particular, we indicate several sub-groups or sub-rings. Among sub-groups, we consider the group of Riordan matrices and indicate its several sub-groups. We show a variety of examples (approximately 20) of matrices that are composed of the sequences of important polynomial or number families as entries of certain lower-triangular infinite matrices. New, significant relationships between these families can be discovered by applying well-known matrix operations like multiplication and inverse calculation to this representation. The paper intends to compile numerous simple facts about the lower-triangular matrices, specifically the family of Rionian matrices, and briefly review their properties.

A π-quasi-upper triangular (resp. π-quasi-lower triangular ) π-equitable matrix is a π-equitable matrix whose quotient matrix is upper triangular (resp. lower triangular). A π-quasi-diagonal π-equitable matrix is a π-equitable matrix whose quotient matrix is diagonal. Given matrices A, B ∈ Mn, we say that A and B are similar in a subset S ⊆ Mn if B = P −1AP for some invertible P ∈ S .Let π be a partition of n, and let A ∈ Mπ . We develop an efficient method for constructing a π-quasi-upper triangular matrix similar to A within Mπ . When the quotient matrix of A is diagonalizable, we further obtain a π-quasi-diagonal matrix similar to A in the same ring.The method is efficient because the similarity problem for a π-equitable matrix reduces to the much smaller quotient matrix, which can be triangularized or diagonalized directly. The resulting structure is then lifted to a corresponding similarity transformation in Mπ , yielding a quasi-triangular or quasi-diagonal form without ever leaving the class of π- equitable matrices.

We present an algebraic framework for constructing challenge–response authentication protocols based on powers of non-diagonalizable matrices over finite fields. The construction relies on upper triangular Toeplitz matrices with a single Jordan block and on their structured power expansions, which induce nonlinear relations between matrix parameters and exponents through an autopotency phenomenon. The protocol is built from a cyclic family of matrix products derived from secret matrices (Ai)i=1n⊂GLk(Fp): for each index i, a product Pi=AiAi+1…Ai+n−1 is formed (indices modulo n), and its power Pi(x) is published for a secret exponent x. The resulting family of powered products is linked by conjugation via the unknown factors Ai, enabling an interactive authentication mechanism in which the prover demonstrates the knowledge of selected factors by satisfying explicit conjugacy relations. We formalize the underlying algebraic problems in terms of factor recovery and conjugacy identification from powered products, and analyze how the enforced non-diagonalizable structure and Toeplitz constraints lead to coupled multivariate polynomial systems. These systems arise naturally from the algebraic design of the construction and do not admit immediate reductions to classical discrete logarithm settings. The framework illustrates how non-diagonalizable matrix structures and structured conjugacy relations can be used to define concrete authentication primitives in noncommutative algebraic settings, and provides a basis for further cryptanalytic and cryptographic investigation.

Let p 0 , p 1 < p 2 < ⋯ < p n be positive real numbers and r a real number. The Loewner and Kraus matrices associated with the function x r are given by L r = [ p i r − p j r p i − p j ] and K r = [ 1 p i − p j ( p i r − p 0 r p i − p 0 − p j r − p 0 r p j − p 0 ) ] respectively. Inertia of these matrices have been computed in Bhatia et al. [Inertia of loewner matrices. Indiana Univ Math J. 2016;65(4):1251–1261] and Sano and Takeuchi [Inertia of Kraus matrices. J Spectr Theory. 2022;12:1443–1457] separately. The inertia of the matrix K r is same as that of inertia of the matrix L r − 1 , when r>0 and inertia of the matrix K r is same as that of inertia of the matrix L − r , when r ≤ 0. In this paper, a congruence relation has been established between these matrix families using an upper triangular matrix. Motivated from this, a general characterization for symmetric matrices to be congruent using upper triangular matrix has been provided.

Let T = ( A 0 U B ) be a triangular matrix ring with A and B two rings, and B U A a ( B , A ) -bimodule. We first study left strongly cotorsion modules over T. After that, the strongly cotorsion dimension for left T-modules and the left global strongly cotorsion dimension of T are studied. Under mild assumptions, we provide certain bounds for the strongly cotorsion dimension of left T-modules and the left global strongly cotorsion dimension of T. Finally, by using the technique of duality pairs, the right strongly torsion-free T-modules and related dimensions are also discussed.

A bstract E.B. Vinberg developed a theory of homogeneous convex cones $$C\subset V={\mathbb{R}}^{n}$$ , which has many applications. He gave a construction of such cones in terms of non-associative rank n matrix T-algebras $$\mathcal{T}$$ , that consist of vector-valued n × n matrices X = || x ij ||, x ij ∈ V ij where V ij are Euclidean vector spaces. The multiplication in a T-algebra is determined by a system of isometric maps V ij × V jk → V ik , s.t. | v ij · v jk | = | v ij | · | v jk | that satisfies some axioms. A T-algebra is determined by its associative subalgebra of upper triangular matrices $$\mathcal{G}$$ or its niladical $$\mathcal{N}$$ , called the Nil-algebra. The connected Lie group $$G\subset \mathcal{G}$$ of the upper triangular (non-degenerate) matrices acts in the vector space $$Her{m}_{n}\subset \mathcal{T}$$ of Hermitian matrices and the orbit C = G ( I ) ⊂ Herm n of the identity matrix I is a convex cone with a simply transitive action of G . Conversely, any homogeneous convex cone is obtained by this construction. Generalizing the notion of rank 3 Clifford T-algebra [1, 2], we define notions of rank n special T-algebra and Clifford Nil-algebra, which define a special Vinberg cone. We associate with a Clifford Nil-algebra $$\mathcal{N}$$ a directed acyclic graph $$\Gamma =\Gamma (\mathcal{N})$$ of diameter 1 and show that Clifford Nil-algebras with given graph Γ bijectively correspond to its admissible equipments. This gives an effective method of classification of Clifford Nil-algebras and associated special Vinberg cones. We apply this approach for explicit classification of rank 4 special Vinberg cones.