Determinan Matriks Sirkulan Dengan Metode Kondensasi Dodgson

One of the important topics in mathematics is matrix theory. There are various types of matrix, one of which is a circulant matrix. Circulant matrix generally fulfill the same operating axioms as square matrix, except that there are some specific properties for the circulant matrix. Every square matrix has a determinant. The concept of determinants is very useful in the development of mathematics and across disciplines. One method of determining the determinant is condensation. The condensation method is classified as a method that is not widely known. The condensation matrix method in determining the determinant was proposed by several scientists, one of which was Charles Lutwidge Dodgson with the Dodgson condensation method. This paper will discuss the Dodgson condensation method in determining the determinant of the circulant matrix. The result of the condensation of the matrix will affect the size of the original matrix as well as new matrix entries. Changes in the circulant matrix after Dodgson's conduction load the Toeplitz matrix, in certain cases, the determinant of the circulant matrix can also be determined by simple mental computation.

Hankel determinants of linear combinations of moments of orthogonal polynomials, II

We present a formula that expresses the Hankel determinants of a linear combination of length $$d+1$$ d + 1 of moments of orthogonal polynomials in terms of a $$d\times d$$ d × d determinant of the orthogonal polynomials. This formula exists somehow hidden in the folklore of the theory of orthogonal polynomials but deserves to be better known, and be presented correctly and with full proof. We present four fundamentally different proofs, one that uses classical formulae from the theory of orthogonal polynomials, one that uses a vanishing argument and is due to Elouafi (J Math Anal Appl 431:1253–1274, 2015) (but given in an incomplete form there), one that is inspired by random matrix theory and is due to Brézin and Hikami (Commun Math Phys 214:111–135, 2000), and one that uses (Dodgson) condensation. We give two applications of the formula. In the first application, we explain how to compute such Hankel determinants in a singular case. The second application concerns the linear recurrence of such Hankel determinants for a certain class of moments that covers numerous classical combinatorial sequences, including Catalan numbers, Motzkin numbers, central binomial coefficients, central trinomial coefficients, central Delannoy numbers, Schröder numbers, Riordan numbers, and Fine numbers.

Determinan Matriks FLScircr Bentuk Khusus n×n,n≥3 Menggunakan Metode Salihu

Determinan mempunyai peranan penting dalam menyelesaikan beberapa persoalan dalam matriks dan banyak dipergunakan dalam ilmu matematika maupun ilmu terapannya. Kondensasi Salihu merupakan salah satu metode yang dapat digunakan dalam menentukan determinan matriks yang memiliki ordo . Metode Kondensasi Salihu merupakan metode gabungan antara Kondensasi Dodgson dan Kondensasi Chio. Penelitian ini bertujuan untuk menentukan determinan matriks bentuk khusus dengan menggunakan metode Kondensasi Salihu. Dalam menentukan determinan matriks bentuk khusus terdapat beberapa langkah yang dikerjakan. Pertama diperhatikan pola detrminan matriks bentuk khusus berode sampai . Kedua pembuktian bentuk umum determinan menggunakan metode induksi matematika. Hasil yang diperoleh adalah didapatkannya bentuk umum dari matriks bentuk khusus. Aplikasi juga dibahas didalam bentuk contoh. [Determinants have an important role in solving several problems in the matrics and are widely used in mathematics and applied sciences. Salihu condensation is one method that can be used to determine the determinant of a matrics that has an order The Salihu Condensation Method is a combined method between Dodgson Condensation and Chio Condensation. This study aims to determine the determinant of a specially matrics form using the Salihu Condensation method. In determining the determinant of a specially matrics there are several steps taken. First, attention the determinant pattern of a specially matrics in orde of 3×3 to 10×10. Second, prove of the general form of determinant using the mathematical induction method. The result obtained is the determination of the general determinant from of a specially matrics.]

Almost Product Evaluation of Hankel Determinants

An extensive literature exists describing various techniques for the evaluation of Hankel determinants. The prevailing methods such as Dodgson condensation, continued fraction expansion, LU decomposition, all produce product formulas when they are applicable. We mention the classic case of the Hankel determinants with binomial entries ${3 k +2 \choose k}$ and those with entries ${3 k \choose k}$; both of these classes of Hankel determinants have product form evaluations. The intermediate case, ${3 k +1 \choose k}$ has not been evaluated. There is a good reason for this: these latter determinants do not have product form evaluations. In this paper we evaluate the Hankel determinant of ${3 k +1 \choose k}$. The evaluation is a sum of a small number of products, an almost product. The method actually provides more, and as applications, we present the salient points for the evaluation of a number of other Hankel determinants with polynomial entries, along with product and almost product form evaluations at special points.

"Trivializing'' generalizations of some Izergin-Korepin-type determinants

Combinatorics International audience We generalize (and hence trivialize and routinize) numerous explicit evaluations of determinants and pfaffians due to Kuperberg, as well as a determinant of Tsuchiya. The level of generality of our statements render their proofs easy and routine, by using Dodgson Condensation and/or Krattenthaler's factor exhaustion method.

Dodgson condensation, alternating signs and square ice

Starting with the numbers 1,2,7,42,429,7436, what is the next term in the sequence? This question arose in the area of mathematics called algebraic combinatorics, which deals with the precise counting of sets of objects, but it goes back to Lewis Carroll's work on determinants. The resolution of the problem was only achieved at the end of the last century, and with two completely different approaches: the first involved extensive verification by computer algebra and a huge posse of referees, while the second relied on an unexpected connection with the theory of ‘square ice’ in statistical physics. This paper, aimed at a general scientific audience, explains the background to this problem and how subsequent developments are leading to a fruitful interplay between algebraic combinatorics, mathematical physics and number theory.