### A note on the inversion of Sylvester matrices in control systems.

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Recently Prodinger [8] considered the reciprocal super Catalan matrix and gave explicit formulæ for its LU-decomposition, the LU-decomposition of its inverse, and obtained some related matrices. For all results, q-analogues were also presented. In this paper, we define and study a variant of the reciprocal super Catalan matrix with two additional parameters. Explicit formulæ for its LU-decomposition, LUdecomposition of its inverse and the Cholesky decomposition are obtained. For all results, q-analogues...

The paper presents an iterative algorithm for computing the maximum cycle mean (or eigenvalue) of $n\times n$ triangular Toeplitz matrix in max-plus algebra. The problem is solved by an iterative algorithm which is applied to special cycles. These cycles of triangular Toeplitz matrices are characterized by sub-partitions of $n-1$.

It is shown that a real Hankel matrix admits an approximate block diagonalization in which the successive transformation matrices are upper triangular Toeplitz matrices. The structure of this factorization was first fully discussed in [1]. This approach is extended to obtain the quotients and the remainders appearing in the Euclidean algorithm applied to two polynomials u(x) and v(x) of degree n and m, respectively, whith m < ...

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.

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...

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.

Cet article présente trois résultats distincts. Dans une première partie nous donnons l’asymptotique quand $N$ tend vers l’infini des coefficients des polynômes orthogonaux de degré $N$ associés au poids ${\varphi}_{\alpha}\left(\theta \right)={|1-{e}^{i\theta}|}^{2\alpha}{f}_{1}\left({e}^{i\theta}\right)$, où ${f}_{1}$ est une fonction strictement positive suffisamment régulière et $\alpha \>\frac{1}{2},\phantom{\rule{1em}{0ex}}\alpha \in \mathbb{R}\setminus \mathbb{N}$. Nous en déduisons l’asymptotique des éléments de l’inverse de la matrice de Toeplitz ${T}_{N}\left({\varphi}_{\alpha}\right)$ au moyen d’un noyau intégral ${G}_{\alpha}.$ Nous prolongeons ensuite un résultat de A. Böttcher et H. Windom relatif à l’asymptotique de la valeur propre...

Given a sequence of real or complex numbers, we construct a sequence of nested, symmetric matrices. We determine the $LU$- and $QR$-factorizations, the determinant and the principal minors for such a matrix. When the sequence is real, positive and strictly increasing, the matrices are strictly positive, inverse $M$-matrices with symmetric, irreducible, tridiagonal inverses.

For p ≡ 1 (mod 4), we prove the formula (conjectured by R. Chapman) for the determinant of the (p+1)/2 × (p+1)/2 matrix $C=\left({C}_{ij}\right)$ with ${C}_{ij}=((j-i)/p)$.

Recently, determinant computation of circulant type matrices with well-known number sequences has been studied, extensively. This study provides the determinants of the RFMLR, RLMFL, RFPrLrR and RLPrFrL circulant matrices with generalized number sequences of second order.

In this paper, we obtain an eigenvalue decomposition for any complex skew-persymmetric anti-tridiagonal Hankel matrix where the eigenvector matrix is orthogonal.