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Descriptor fractional linear systems with regular pencils

Tadeusz Kaczorek (2013)

International Journal of Applied Mathematics and Computer Science

Methods for finding solutions of the state equations of descriptor fractional discrete-time and continuous-time linear systems with regular pencils are proposed. The derivation of the solution formulas is based on the application of the Z transform, the Laplace transform and the convolution theorems. Procedures for computation of the transition matrices are proposed. The efficiency of the proposed methods is demonstrated on simple numerical examples.

Determinant evaluations for binary circulant matrices

Christos Kravvaritis (2014)

Special Matrices

Determinant formulas for special binary circulant matrices are derived and a new open problem regarding the possible determinant values of these specific circulant matrices is stated. The ideas used for the proofs can be utilized to obtain more determinant formulas for other binary circulant matrices, too. The superiority of the proposed approach over the standard method for calculating the determinant of a general circulant matrix is demonstrated.

Determinant Representations of Sequences: A Survey

A. R. Moghaddamfar, S. Navid Salehy, S. Nima Salehy (2014)

Special Matrices

This is a survey of recent results concerning (integer) matrices whose leading principal minors are well-known sequences such as Fibonacci, Lucas, Jacobsthal and Pell (sub)sequences. There are different ways for constructing such matrices. Some of these matrices are constructed by homogeneous or nonhomogeneous recurrence relations, and others are constructed by convolution of two sequences. In this article, we will illustrate the idea of these methods by constructing some integer matrices of this...

Determinants and inverses of circulant matrices with complex Fibonacci numbers

Ercan Altınışık, N. Feyza Yalçın, Şerife Büyükköse (2015)

Special Matrices

Let ℱn = circ (︀F*1 , F*2, . . . , F*n︀ be the n×n circulant matrix associated with complex Fibonacci numbers F*1, F*2, . . . , F*n. In the present paper we calculate the determinant of ℱn in terms of complex Fibonacci numbers. Furthermore, we show that ℱn is invertible and obtain the entries of the inverse of ℱn in terms of complex Fibonacci numbers.

Déterminants et intégrales de Fresnel

Yves Colin de Verdière (1999)

Annales de l'institut Fourier

On présente ici une approche directe et géométrique pour le calcul des déterminants d’opérateurs de type Schrödinger sur un graphe fini. Du calcul de l’intégrale de Fresnel associée, on déduit le déterminant. Le calcul des intégrales de Fresnel est grandement facilité par l’utilisation simultanée du théorème de Fubini et d’une version linéaire du calcul symbolique des opérateurs intégraux de Fourier. On obtient de façon directe une formule générale exprimant le déterminant en terme des conditions...

Determinants of (–1,1)-matrices of the skew-symmetric type: a cocyclic approach

Víctor Álvarez, José Andrés Armario, María Dolores Frau, Félix Gudiel (2015)

Open Mathematics

An n by n skew-symmetric type (-1; 1)-matrix K =[ki;j ] has 1’s on the main diagonal and ±1’s elsewhere with ki;j =-kj;i . The largest possible determinant of such a matrix K is an interesting problem. The literature is extensive for n ≡ 0 mod 4 (skew-Hadamard matrices), but for n ≡ 2 mod 4 there are few results known for this question. In this paper we approach this problem constructing cocyclic matrices over the dihedral group of 2t elements, for t odd, which are equivalent to (-1; 1)-matrices...

Determinants of matrices associated with incidence functions on posets

Shaofang Hong, Qi Sun (2004)

Czechoslovak Mathematical Journal

Let S = { x 1 , ⋯ , x n } be a finite subset of a partially ordered set P . Let f be an incidence function of P . Let [ f ( x i ∧ x j ) ] denote the n × n matrix having f evaluated at the meet x i ∧ x j of x i and x j as its i , j -entry and [ f ( x i ∨ x j ) ] denote the n × n matrix having f evaluated at the join x i ∨ x j of x i and x j as its i , j -entry. The set S is said to be meet-closed if x i ∧ x j ∈ S for all 1 ≤ i , j ≤ n . In this paper we get explicit combinatorial formulas for the determinants of matrices [ f ( x i ∧ x j ) ] and [ f ( x i ∨ x j ) ] on any meet-closed set S . We also obtain necessary and sufficient conditions for the matrices...

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