Dr. Iyad Abu-Jeib's Research Activities and Research Interests
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Click here for copies of my papers.

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Click here for some of the citations of my papers in books, in wikipedia, in papers and in conferences.

Research Interests:

Pattern Recognition

Algorithms

Theory of Computation

Discrete Mathematics

Matrix Theory

Numerical Linear Algebra

Numerical Analysis

Operator Theory

Frames in Hilbert Space

Summaries of Some of My Publications

Here are summaries of some of my publications that appeared in refereed (peer-reviewed) journals:

Remarks:

  1. The papers that did not appear and will not be appearing soon are not listed here.

  2. In all oy my papers, a pure imaginary number refers to a number of the form \bgroup\color{blue}$ bi$\egroup, where \bgroup\color{blue}$ b$\egroup is any real number (including zero). Thus, in our definition of pure imaginary, we do not exclude the number zero. In other words, we consider 0 a pure imaginary number.

Here are the summaries:

  1. On Matrix $ I^{(-1)}$ of Sinc Methods (a joint paper with Dr. Thomas Shores). In this paper, we study properties of matrix $ I^{(-1)}$ of sinc methods, which is defined as follows:

    Definition 0.1   $ I^{(-1)}$ is the $ n\times n $ matrix defined as follows:

    $\displaystyle I^{(-1)}=\left[ \eta _{ij}\right] _{i,j=1}^{n}$

    where $ \eta _{ij}=e_{i-j} $, $ \displaystyle e_{k}=\frac{1}{2}+s_{k} $, and $ \displaystyle s_{k}=\int _{0}^{k}\operatorname {sinc}(x)dx. $

    Sinc methods are a family of formulas based on the sinc function which give accurate approximations of derivatives and definite and indefinite integrals and convolutions. These methods were developed by Frank Stenger. One of the nice properties of these methods is that they can handle boundary layer problems, integrals with infinite intervals or with singular integrands, and ODEs or PDEs that have coefficients with singularities.

    In this paper, we study properties of this Toeplitz matrix $ I^{(-1)}$. This matrix and its properties are very important in the theory of Sinc indefinite integration and Sinc convolution

    Here is a copy of the paper

  2. Rank-one Perturbations and Transformations of Centrosymmetric Matrices. In this paper, we study the effect of transformations and rank-one perturbations of centrosymmetric matrices on the eigenvalues, eigenvectors, determinants, and inverses.

    Here is a copy of the paper

  3. Centrosymmetric Matrices: Properties and an Alternative Approach. In this paper, we describe a different approach of looking at and handling centrosymmetric matrices. This approach can be used as an alternative method to derive most of the known results about centrosymmetric matrices and new ones. We also identify orthogonal transformations between centrosymmetric matrices and skew-centrosymmetric matrices. One of these transformations is very helpful for reducing centrosymmetric (resp. skew-centrosymmetric) problems to skew-centrosymmetric (resp. centrosymmetric) problems. For example, we can transform every skew-centrosymmetric singular value/determinant problem of even order to a centrosymmetric singular value/determinant problem of even order and vice versa. Moreover, we can transform every linear system in which the matrix of coefficients is centrosymmetric of even order to a linear system in which the matrix of coefficients is skew-centrosymmetric of even order, and vice versa. We also reveal properties for centrosymmetric matrices and skew-centrosymmetric matrices. In addition, we study a new charactarization of centrosymmetric matrices and skew-centrosymmetric matrices.

    Here is a copy of the paper

  4. Centrosymmetric and Skew-centrosymmetric Matrices and Regular Magic Squares. In this paper, we reveal new properties of centrosymmetric and skew-centrosymmetric matrices. We also study properties of structured matrices involving these two types of matrices. For example, we study properties (determinants, eigen structure, singular values, etc) of structured complex matrices that invlove centrosymmetric and skew-centrosymmetric matrices. Hermitian persymmetric matrices are special cases of the matrices we study (which implies Goldstein reduction theorem for Herimitian persymmetric matrices follows as a corollary from our results). As another example, we study properties of regular magic squares and present another proof for the singularity of regular magic squares of even order. We also study singular values of centrosymmetric matrices and skew-centrosymmetric matrices, and mention some of their transformations. Although it is easy to see that the most known property that characterizes the eigen structure of centrosymmetric does not hold for skew-centrosymmetric matrices, we study a summetric-skewsymemtric eigenvector property for the special case when the matrix is also real and skew-symmetric.

    Here is a copy of the paper

  5. On the Counteridentity Matrix. Note that the counteridentity matrix is also called the exchange matrix, the flip matrix, the anti-identity matrix, and the contra-identity matrix. In this paper, we make a comparison between the identity matrix and the counteridentity matrix (aka flip matrix, exchange matrix, contra identity, anti-identity). By the main counterdiagonal of a square matrix, we mean the positions which proceed diagonally from the last entry in the first row to the first entry in the last row. The main counterdiagonal is sometimes called the secondary diagonal or the main anti-diagonal. We will simply say counterdiagonal when we refer to the main counterdiagonal. The counteridentity matrix, denoted $ J$, is the matrix whose elements are all equal to zero except those on the counterdiagonal, which are all equal to 1. Our paper reveals a structured family of matrices with the following property: if $ a + bi$ is an eigenvalue of a matrix of this family, then either $ a = 0$ or $ b = 0$, which means its eigenvalues are either real or pure imaginary. It also describes the eigenvalues of matrices whose entries are all zeros except possibly those on the main diagonal or the main counterdiagonal. In other words, if $ A=(a_{ij})$ is such a matrix ($ A$ is $ n\times n $), then $ a_{ij} = 0$ if $ j \neq i$ and $ j \neq n-i+1$. We construct an analytic homotopy $ H(t)$ in the space of diagonalizable matrices, between the counteridentity and any real skew-symmetric skew-centrosymmetric matrix such that $ H(t)$ has only real or pure imaginary eigenvalues for $ 0\leq t\leq 1$. We study similartites and differences between the identity and the counteridentity.

    Here is a copy of the paper

  6. Classic two-step Durbin-type and Levinson-type algorithms for skew-symmetric Toeplitz matrices. We present fast O(N2) two-step algorithms for solving linear systems of equations involving skew-symmetric Toeplitz matrices. Our approach uses similar approaches to those used by Durbin and Levinson for symmetric Toeplitz matrices but with some tricks to overcome the problem of the singularity of skew-symmetric Toeplitz matrices of odd-order. In the paper, we explain how to derive the algorithm, then we present the algorithm, then we discuss the time-complexity of the algorithm, then we present examples, and finally we present an Octave (a MATLAB-like programming language) program that solves fast any linear system in which the matrix of coefficients is skew-symmeric Toeplitz.

    Here is a copy of the paper

  7. A classic Trench-type algorithm for skew-symmetric Toeplitz matrices. We present fast O(N2 ) two-step algorithm for inverting non-singular skew-symmetric Toeplitz matrices. Our approach uses a similar approach to that used by Trench for symmetric Toeplitz matrices but with some tricks to overcome the problem of the singularity of skew-symmetric Toeplitz matrices of odd-order. In the paper, we explain how to derive the algorithm, then we present the algorithm, then we discuss the time-complexity of the algorithm, then we present an example, and finally we present an Octave program that inverts fast any non-singular skew-symmeric Toeplitz matrix.

    Here is a copy of the paper

    We note that the approcahes we used in the above two papers are completely different than those used by Georg Heinig and Karla Rost for Toeplitz matrices. Our approaches were focused on generalizing the classic Durbin, Levinson, and Trench algorithms for the symnmetric case to the skew-symmetric case. At the time when my papers were written (at first, they were submitted to a journal that took it a long time to referee them, so I withdrew them and submitted them to another journal), I think the only papers for Heinig and Rost about skew-symmetric Toeplitz matrices that were published are those I cited in my paper "Classic Two-step Durbin-Type and Levinson-Type Algorithms for Skew-symmetric Toeplitz Matrices" which is cited in my paper "A Classic Trench-Type Algorithm for Skew-symmetric Toeplitz Matrices". I sent my work above to Heinig who sent me also one of his papers and a few months later another paper.

  8. Algorithms for Centrosymmetric and Skew-centrosymmetric Matrices.

    Here is a copy of the paper.

  9. The Determinant of the Wheel Graph and Conjectures by Young

    Here is a copy of the paper.

  10. Involutions and Generalized Centrosymmetric and Skew-centrosymmetric Matrices

    Here is a copy of the paper.