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It is known that every singular matrix of a finite degree defined over a field can be written as a product of a finite number of idempotent matrices. Insipired by this result we investigate the same problem for $ \mathbb {N}\times \mathbb {N} $ N × N matrices. It turns out that in this case the analogous theorem does not hold. [ABSTRACT FROM AUTHOR]

A min-plus algebra is a linear algebra over the semiring R_ε', equipped with the operations “"⊕'=min" ” and “⊗=+”. In min-plus algebra, there is the concept of characteristic polynomial obtained from permanent of matrix. Min-plus algebra can be extended to an interval min-plus algebra, which is a set 〖 I(R)〗_ε' equipped with the operations ¯("⊕'" ) and ¯("⊗" ). Matrix over interval min-plus algebra has some special forms, one of which is a triangular matrix. The concept of characteristic polynomial and eigenproblem can be applied to a triangular matrix. There is a more concise formula for solving the eigenproblem of triangular matrix because this matrix is a special form of matrix. This research will discuss the characteristic polynomial and solving eigenproblem of triangular matrix over interval min-plus algebra using its characteristic polynomials. The research method used is a literature study. From the research results, the permanent formula and characteristic polynomial formula of the triangular matrix are obtained. It is also obtained that the smallest corner of the characteristic polynomial is the principal eigenvalue and the vector eigen corresponding to the principal eigenvalue can be obtained through the matrix A_λ. For readers who are interested in this topic, can research about characteristic polynomial and eigenproblem of matrices with other special forms over min-plus interval algebra.

In this paper, we compute paradeterminants and parapermanents of some triangular matrices that give bihyperbolic numbers of the Fibonacci type. Using connections between the paradeterminant of triangular matrix and the lower Hessenberg determinant, we also obtain the general term of these sequences. [ABSTRACT FROM AUTHOR]

Background Determinant and characteristic polynomials are important concepts related to square matrices. Due to the absence of additive inverse in max-plus algebra, the determinant of a matrix over max-plus algebra can be represented by a permanent. In addition, there are several types of square matrices over max-plus algebra, including triangular and diagonal strictly double ℝ -astic matrices. A special formula has been devised to determine the permanent and characteristic max-polynomial of those matrices. Another algebraic structure that is isomorphic with max-plus algebra is min-plus algebra. Methods Min-plus algebra is the algebraic structure of triple ( ℝ ε ′ , ⊕ ′ , ⊗ ) . Furthermore, square matrices over min plus algebra are defined by the set of matrices sized n × n , the entries of which are the elements of ℝ ε ′ . Because these two algebraic structures are isomorphic, the permanent and characteristic min-polynomial can also be determined for each square matrix over min-plus algebra, as well as the types of matrices. Results In this paper, we find out the special formulas for determining the permanent and characteristic min-polynomial of the triangular matrix and the diagonal strictly double ℝ -astic matrix. Conclusions We show that the formula for determining the characteristic min-polynomial of the two matrices is the same, for each triangular matrix and strictly double ℝ -astic matrix A , χ A ( x ) = ⨁ r = 0 , 1 , … , n ′ δ n − r ⊗ n r .

The area of matrix functions has received growing interest for a long period of time due to their growing applications. Having a numerical algorithm for a matrix function, the ideal situation is to have an estimate or bound for the error returned alongside the solution. Condition numbers serve this purpose; they measure the first-order sensitivity of matrix functions to perturbations in the input data. We have observed that the existing unstructured condition number leads most of the time to inferior bounds of relative forward errors for some matrix functions at triangular and quasi-triangular matrices. We propose a condition number of matrix functions exploiting the structure of triangular and quasi-triangular matrices. We then adapt an existing algorithm for exact computation of the unstructured condition number to an algorithm for exact evaluation of the structured condition number. Although these algorithms are direct rather than iterative and useful for testing the numerical stability of numerical algorithms, they are less practical for relatively large problems. Therefore, we use an implicit power method approach to estimate the structured condition number. Our numerical experiments show that the structured condition number captures the behavior of the numerical algorithms and provides sharp bounds for the relative forward errors. In addition, the experiment indicates that the power method algorithm is reliable to estimate the structured condition number. [ABSTRACT FROM AUTHOR]

Broadband filtering and reconstruction-based spectral measurement represent a hot technical route for miniaturized spectral measurement; the measurement encoding scheme has a great effect on the spectral reconstruction fidelity. The existing spectral encoding schemes are usually complex and hard to implement; thus, the applications are severely limited. Considering this, here, a simple spectral encoding method based on a triangular matrix is designed. The condition number of the proposed spectral encoding system is estimated and demonstrated to be relatively low theoretically; then, verification experiments are carried out, and the results show that the proposed encoding can work well under precise or unprecise encoding and measurement conditions; therefore, the proposed scheme is demonstrated to be an effective trade-off of the spectral encoding efficiency and implementation cost. [ABSTRACT FROM AUTHOR]

Health insurance company have to determine claim reserve that’s suitable with the existing condition. There is three party that’s involved in the health insurance management, namely the policy holder, Admedika as the third party administration, and also the insurance company itself as the (insurer). When the policy holder obtained treatments whose financing is done through a health insurance, then the health insurance company have the obligation to finish the financial matters. Delays in payments from insurance companies to health facilities are caused, among others, by the administrative process. Thus, every claim submitted by the insured party to the insurance company will be settled in stages to the health facilities. The data presented from these conditions form a triangle matrix (run-off triangle) which then becomes the basis for estimating the amount of IBNR claims reserves. The Bornhuetter-Ferguson (BF) method involves the amount of premium that has become income for the company and calculates the Ultimate Claim value in estimating the amount of claim reserves. This method is the result of the development of the previous method, Chain-Ladder (CL), which only relies on historical data on claim payments. Premium calculations need to be involved in health insurance, because the insurance period is short, which is only one year. Insurance companies haven't had time to turn around the money to invest, so payment of claims will depend more on the premium that becomes income for the company (earned premium). The estimated claim reserve value is more suitable and robust than the CL method. Estimated claim reserves that occur in the 2nd event period amount to IDR50,658,714 with an estimated interval for the 2nd event period between IDR10,215,477 and IDR91,101,950

Let D be an infinite division ring with centre F and T (∞ , D) the group of infinite upper triangular invertible matrices indexed by N × N over D. In this paper, we first show that if D is finite dimensional over F, then every element in T (∞ , D) whose diagonal entries are commutators of D ∖ { 0 } is a commutator of an infinite diagonal matrix and another infinite upper triangular matrix. Some applications are also shown. For example, if dim F ⁡ D ≤ 4 , then every element in the commutator subgroup [ G L V K (∞ , D) , G L V K (∞ , D) ] of the Vershik-Kerov group G L V K (∞ , D) is a commutator. [ABSTRACT FROM AUTHOR]

Broadband filtering and reconstruction-based spectral measurement represent a hot technical route for miniaturized spectral measurement; the measurement encoding scheme has a great effect on the spectral reconstruction fidelity. The existing spectral encoding schemes are usually complex and hard to implement; thus, the applications are severely limited. Considering this, here, a simple spectral encoding method based on a triangular matrix is designed. The condition number of the proposed spectral encoding system is estimated and demonstrated to be relatively low theoretically; then, verification experiments are carried out, and the results show that the proposed encoding can work well under precise or unprecise encoding and measurement conditions; therefore, the proposed scheme is demonstrated to be an effective trade-off of the spectral encoding efficiency and implementation cost.

In this work, we explore an extension of the Omega calculus in the context of matrix analysis introduced recently by Neto [Matrix analysis and Omega calculus. SIAM Rev. 2020;62(1):264–280]. We obtain Omega representations of analytic functions of three important classes of matrices: companion, tridiagonal, and triangular. Our representation recovers the main results of Chen and Louck [The combinatorial power of the companion matrix. Linear Algebra Appl. 1996;232:261–278] on the powers of the companion matrix. Furthermore, we generalize previous work on the powers of tridiagonal matrices due to Gutiérrez–Gutiérrez in [Powers of tridiagonal matrices with constant diagonals. Appl Math Comput. 2008;206(2):885–891], Öteleş and Akbulak [Positive integer powers of certain complex tridiagonal matrices. Appl Math Comput. 2013;219(21):10448–10455], and triangular matrices following Shur [A simple closed form for triangular matrix powers. Electron J Linear Algebra. 2011;22:1000–1003]. [ABSTRACT FROM AUTHOR]

Let L be the infinite lower triangular Toeplitz matrix with first column (µ, a1, a2, ..., ap, a1, ..., ap, ...)T and let D be the infinite diagonal matrix whose entries are 1, 2, 3, . . . Let A := L + D be the sum of these two matrices. Bünger and Rump have shown that if p = 2 and certain linear inequalities between the parameters µ, a1, a2, are satisfied, then the singular values of any finite left upper square submatrix of A can be bounded from below by an expression depending only on those parameters, but not on the matrix size. By extending parts of their reasoning, we show that a similar behaviour should be expected for arbitrary p and a much larger range of values for µ, a1, ..., ap. It depends on the asymptotics in µ of the l2-norm of certain sequences defined by linear recurrences, in which these parameters enter. We also consider the relevance of the results in a numerical analysis setting and moreover a few selected numerical experiments are presented in order to show that our bounds are accurate in practical computations.

In this paper, characterizations of right core inverse are given by one-sided invertibility. The necessary and sufficient conditions, which guarantee that paq has right core inverses, are investigated. Furthermore, characterizations of right core inverses of triangular matrices, 2 × 2 matrices and a companion matrix are considered. [ABSTRACT FROM AUTHOR]

Let ℜ be a ring with identity 1 whose tripotents are only -1, 0, and 1. It is characterized the structure of tripotents in T (ℜ) which is the ring of triangular matrices over ℜ. In addition, when ℜ is finite, it is given number of the tripotents in Tn(ℜ) which is the ring of n x n dimensional triangular matrices over ℜ with n being a positive integer. [ABSTRACT FROM AUTHOR]

Due to the robustness, the matrix inversion methods based on matrix factorization are often adopted in engineering while the triangular matrix inversion is one key part of those methods. This brief reviews the adjoint matrix and presents a novel scheme for field-programmable gate arrays (FPGAs) calculating the triangular matrix inversion. Employing the more characteristics of both the triangular matrix and its inversion, the proposed diagonal-wise algorithm for the triangular matrix inversion has the high parallelism and extensibility in the hardware implementation and is suitable for the different matrix triangular factorization (QR, LDL, Cholesky and LU). Meanwhile, the recursive diagonal-wise algorithm is designed for the large scale triangular matrices. Compared with the traditional row-/column-wise methods, our algorithm has a good performance at the low computation load. Finally, the experiments are conducted on one Xilinx Virtex-7 FPGA to illustrate the performances of the four different methods. [ABSTRACT FROM AUTHOR]

Abstract In this article, we study some Riemann–Hilbert problems with shift on the Lyapunov curve for generalized polyanalytic functions, which are null-solutions of a class of iterated Beltrami equations. Firstly, we obtain two integral representations of these functions by using Cauchy formula associated with the Beltrami equations and explicitly constructing two weakly singular kernels. Next, we develop a theory of matrix factorization for triangular matrix functions with shift in the frame of Beltrami equations and solve the Riemann–Hilbert problems by using them and the decomposition theorem for the iterated Beltrami equations. Finally, we get explicit formulae of solutions and conditions of solvability for the Riemann–Hilbert problems.

Due to the increase in the use of precision agriculture, field trials have increased in size to allow for genomic selection tool development by linking quantitative phenotypic traits to sequence variations in the DNA of various crops. Crop row detection is an important step to enable the development of an efficient downstream analysis pipeline for genomic selection. In this paper, an efficient crop row detection algorithm was proposed that detected crop rows in colour images without the use of templates and most other pre-information such as number of rows and spacing between rows. The method only requires input on field weed intensity. The algorithm was robust in challenging field trial conditions such as variable light, sudden shadows, poor illumination, presence of weeds and noise and irregular crop shape. The algorithm can be applied to crop images taken from the top and side views. The algorithm was tested on a public dataset with side view images of crop rows and on Genomic Sub-Selection dataset in which images were taken from the top view. Different analyses were performed to check the robustness of the algorithm and to the best of authors' knowledge, the Receiver Operating Characteristic graph has been applied for the first time in crop row detection algorithm testing. Lastly, comparing this algorithm with several state-of-the-art methods, it exhibited superior performance. [ABSTRACT FROM AUTHOR]

Rhoades and Savaş [6], [11] established necessary conditions for inclusions of the absolute matrix summabilities under additional conditions. In this paper, we determine necessary or sufficient conditions for some classes of infinite matrices, and using this, we get necessary or sufficient conditions for more general absolute summabilities applied to all matrices. [ABSTRACT FROM AUTHOR]

This brief briefly addresses the problem of power flow solution for direct-current (dc) networks with radial configuration and constant power loads (CPLs). It proposes a novel iterative method based on the upper triangular relationship between nodal and branch currents, it also uses a primitive impedance matrix. The main advantage of this method lies in the possibility of avoiding inversions of non-diagonal matrices, which allows its convergence to be improved in terms of the number of iterations and processing times required in comparison to classical admittance-based methods. Three different radial dc resistive networks composed by 21, 33, and 69 nodes are employed to validate the effectiveness of the proposed power flow solution method. For comparison purposes, the Newton–Raphson method, and also successive approximations and Taylor-based approaches are implemented. All simulations have performed in MATLAB software. [ABSTRACT FROM AUTHOR]

Let n ⩾ 2 be an integer and let F be a field with F ⩾ 3. Let T n (F) be the ring of n × n upper triangular matrices over F with centre Z . Fixing an integer 2 ⩽ k ⩽ n , we prove that an additive map ψ : T n (F) → T n (F) satisfies A ψ (A) = ψ (A) A for all rank k matrices A ∈ T n (F) if and only if there exist an additive map μ : T n (F) → Z , Z ∈ Z and α ∈ F in which α = 0 when F > 3 or k

We obtained several equivalent conditions for the existence of core inverses and dual core inverses of triangular matrices over a ring with involution. As applications, some necessary and sufficient conditions for the (2,2,0) core inverse problem are given. [ABSTRACT FROM AUTHOR]

In the paper, the authors establish two identities to express higher order derivatives and integer powers of the generating function of the Chebyshev polynomials of the second kind in terms of integer powers and higher order derivatives of the generating function of the Chebyshev polynomials of the second kind respectively, find an explicit formula and an identity for the Chebyshev polynomials of the second kind, conclude the inverse of an integer, unit, and lower triangular matrix, derive an inversion theorem, present several identities of the Catalan numbers, and give some remarks on the closely related results including connections of the Catalan numbers with the Chebyshev polynomials of the second kind, the central Delannoy numbers, and the Fibonacci polynomials respectively. [ABSTRACT FROM AUTHOR]

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Let End k ⁢ (V) {\mathrm{End}_{k}(V)} denote the ring of all linear transformations of an arbitrary k-vector space V over a field k. We define X ⊆ End k ⁢ (V) {X\subseteq\mathrm{End}_{k}(V)} to be triangularizable if V has a well-ordered basis such that X sends each vector in that basis to the subspace spanned by basis vectors no greater than it. We then show that an arbitrary subset of End k ⁢ (V) {\mathrm{End}_{k}(V)} is strictly triangularizable (defined in the obvious way) if and only if it is topologically nilpotent. This generalizes the theorem of Levitzki that every nilpotent semigroup of matrices is triangularizable. We also give a description of the triangularizable subalgebras of End k ⁢ (V) {\mathrm{End}_{k}(V)} , which generalizes a theorem of McCoy classifying triangularizable algebras of matrices over algebraically closed fields. [ABSTRACT FROM AUTHOR]

Applying the apparatus of triangular matrices, we proved new recurrent formulas for Horadam numbers with even (odd) subscripts through determinants of tridiagonal matrix. [ABSTRACT FROM AUTHOR]