On special matrices related to Cauchy and Toeplitz matrices

Cogent Mathematics, 2016

In this paper, we investigate some properties of Toeplitz matrices with respect to different matrix products. We also give some results regarding circulant matrices, skew-circulant matrices and approximation by Toeplitz matrices over the field of complex numbers.

2020

We present a proof of determinant of special nonsymmetric Toeplitz matrices conjectured by Anelić and Fonseca in <cit.>. A proof is also demonstrated for a more general theorem. The two conjectures are therefore just two possible results, under two specific settings. Numerical examples validating the theorem are provided.

Arabian Journal of Mathematics, 2017

We show that the characteristic polynomial of a symmetric pentadiagonal Toeplitz matrix is the product of two polynomials given explicitly in terms of the Chebyshev polynomials.

Operators and Matrices, 2009

The purpose of this paper is to compute the asymptotics of determinants of finite sections of operators that are trace class perturbations of Toeplitz operators. For example, we consider the asymptotics in the case where the matrices are of the form (a i−j ± a i+j+1−k ) i,j=0...N −1 with k is fixed. We will show that this example as well as some general classes of operators have expansions that are similar to those that appear in the Strong Szegö Limit Theorem. We also obtain exact identitities for some of the determinants that are analogous to the one derived independently by Geronimo and Case and by Borodin and Okounkov for finite Toeplitz matrices. These problems were motivated by considering certain statistical quantities that appear in random matrix theory. * [email protected].

2005

], respectively. Since this paper does not provide any new results, it will not be published anywhere.

1996

In this paper we consider a class of matrices, each of which is the sum of an identity matrix and a self-adjoint block Toeplitz matrix that has a symmetric band o{" zero blocks. A number of general results for block Toeplitz matrices are specialized to this class.

1999

Let T be a skew-symmetric Toeplitz matrix with entries in a ®nite ®eld. For all positive integers n let Tn be the upper n n corner of T, with nullity mn ˆ m Tn†. The sequence fmn : n 2 Ng satis®es a unimodality property and is eventually periodic if the entries of T satisfy a periodicity condition. We compute the maximum value and the period of the nullity sequence for Toeplitz matrices of ®nite bandwidth. This sequence satis®es a certain symmetry condition about its maximal values. These results apply to give some information about the ranks of general skew-symmetric Toeplitz matrices with eventually periodic entries. Ó 1999 Elsevier Science Inc. All rights reserved.

Czechoslovak Mathematical Journal, 2008

Let a, b and c be fixed complex numbers. Let M n (a, b, c) be the n × n Toeplitz matrix all of whose entries above the diagonal are a, all of whose entries below the diagonal are b, and all of whose entries on the diagonal are c. For 1 ⩽ k ⩽ n, each k × k principal minor of M n (a, b, c) has the same value. We find explicit and recursive formulae for the principal minors and the characteristic polynomial of M n (a, b, c). We also show that all complex polynomials in M n (a, b, c) are Toeplitz matrices. In particular, the inverse of M n (a, b, c) is a Toeplitz matrix when it exists.

IEEE Transactions on Signal Processing, 2000

We extend a recurrence relation between determinants of real symmetric Toeplitz matrices to the case where these matrices are singular. The relation exploits the even-odd structure of Toeplitz matrices and contains two factors: one corresponding to the even and the other corresponding to the odd part of their spectrum.

The word "matrix" comes from the Latin word for "womb" because of the way that the matrix acts as a womb for the data that it holds. The first known example of the use matrices was found in a Chinese text called Nine Chapters of the Mathematical Art, which is thought to have originated somewhere between 300 B.C. and 200 A.D. The modern method of matrix solution was developed by a German mathematician and scientist Carl Friedrich Gauss. There are many different types of matrices used in different modern career fields. We introduce and discuss the different types of matrices that play important roles in various fields.