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49
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Introduction
My main research interests are Matrix Analysis and its applications in Electircal Engineering. A large part of my work is on computational methods of generalized inverses of matrices (eg. the Moore-Penrose inverse, the Drazin inverse, the Group inverse, weighted inverses and different types of outer inverses) both numerically and Symbolically. In addition, I am interested in applications of generalized inverses in optimization problems in Finance and in Statistics.
Additional affiliations
September 2014 - August 2015
October 2007 - present
October 2007 - present
Education
January 1998 - June 2006
Publications
Publications (49)
The aim of this paper is threefold. Firstly, we define the necessary quantities associated to the lumpability of a Markov chain and study their fundamental properties. Secondly, we examine the case of approximate lumpability of a non-lumpable Markov and an efficient method of minimizing the error in the approximation. Finally, we introduce a family...
In this work we propose an unbiased estimator for a multiple linear regression model of the CAPM in the presence of multicollinearity in the explanatory variables. Multicollinearity is a common problem in empirical Econometrics. The existing methods so far do not deal with cases of perfect multicollinearity. This new optimization method that belong...
In this paper, novel representations of generalized inverses of rational matrices are developed. Therefore, a unified approach for the computation of {1,2,3} and {1,2,4} inverses and Moore-Penrose inverse of a given matrix A is considered. Full-rank QDR decomposition of a rational matrix is utilized to avoid the square roots of rational expressions...
In the last few years, detection has become a powerful methodology for network pro-
tection and security. This paper presents a new detection scheme for data recorded
over a computer network. This approach is applicable to the broad scientific field
of information security, including intrusion detection and prevention. The proposed
method employ...
In the last few years, detection has become a powerful methodology for network protection and network security measures. This paper presents a new detection scheme for data recorded over a computer network, applicable on the broad scientific field of information security, including intrusion detection and prevention. The proposed method employs bi-...
The aim of this paper is threefold. Firstly, we define the necessary quantities associated to the lumpability of a Markov chain and study their fundamental properties. Secondly, we examine the case of approximate lumpability of a non-lumpable Markov and an efficient method of minimizing the error in the approximation. Finally, we introduce a family...
We are looking for an answer to the next challenging question: is the Newton iterative rule the only iterative method for computing generalized inverses with quadratic convergence? Our answer is that it is possible to find a class of quadratically convergent iterative methods containing the Newton method. To that end, we introduce and investigate a...
In this paper we consider the minimization of a positive semidefinite quadratic form, having a singular corresponding matrix $H$. We state the dual formulation of the original problem and treat both problems only using the vectors $x \in \mathcal{N}(H)^\perp$ instead of the classical approach of convex optimization techniques such as the null space...
In this paper we consider the minimization of a positive semidefi-nite quadratic form, having a singular corresponding matrix H. We state the dual formulation of the original problem and treat both problems only using the vectors x ∈ N (H) ⊥ , instead of the classical approach of convex optimization techiques such as the null space method. Given th...
Following the results of our recent paper, regarding the Aluthge transform of polynomial matrices, the symbolic
computation of the Duggal transform of a polynomial matrix A is developed in this paper, using the polar decomposition and the singular value decomposition of A. Thereat, the polynomial singular value decomposition method is utilized, whi...
Two finite recurrent procedures for computing
and
-inverses of a matrix are presented. Each of introduced methods exploits certain matrix product which includes the Moore–Penrose inverse of a symmetric matrix and a generalization of the Sherman-Morrison formula to the case of the Moore–Penrose inverse of a symmetrically rank one modified matrix. T...
In this work we present some relationships between an EP matrix T, its Aluthge transform ∆(T) or the λ-Aluthge transform ∆ λ (T) and the Moore-Penrose inverse T † . We prove that the λ-Aluthge transform of T is also an EP matrix, and the same thing holds for ∆ λ (T) † and ∆ λ (T † ). Also, we explore the product ∆ λ (T)T, the connections between ∆(...
The algorithm for the symbolic computation of the Aluthge transform of a polynomial matrix is derived in this paper. For this purpose, the well-known PSVD by PQRD algorithm is considered to avoid square roots of polynomials in the Aluthge transform matrix. The algorithm for the symbolic computation of the polar decomposition for polynomial matrices...
A finite recursive procedure for computing {2, 4} generalized inverses and the analogous recursive procedure
for computing {2, 3} generalized inverses of a given complex matrix are presented. The starting points
of both introduced methods are general representations of these classes of generalized inverses. These representations are formed using ce...
The further investigation of the image restoration method introduced in [19, 20] is presented in this paper. Continuing investigations in that area, two additional applications of the method are investigated. More precisely, we consider the possibility to replace the available matrix in the method by the restoration obtained applying the Tikhonov r...
In this work we introduce a new kind of generalized inverse, called the T-restricted weighted Drazin
inverse of A with respect to a positive semidefinite/definite matrix T. The new approach proposed
makes use of the normal Drazin equation of a non consistent matrix equation as the constraint set
of the minimization. In addition, when T is positive se...
In the last few years, watermarking has become a powerful tool for data hiding and copyright protection. This paper presents a new noise-robust scheme for signal watermark embedding and extraction, applicable on the broad scientific field of information security, including speech and audio secure transmission. The proposed method employs bi-dimensi...
We follow the idea to find solution of quadratic minimization problems restricted by linear constraints and to investigate underlying generalized inverses. A generalization of this approach leads to a new kind of outer generalized inverse, called the T -minimal G-constrained inverse of A (called the minimal (T,G) inverse of A shortly), with respect...
In practice, the recorded image unavoidably represents a degraded version of the original scene because of inevitable imperfections in the imaging and capturing process. Medical images, satellite images, astronomical images, or poor-quality family portraits are often blurred. A wide range of different degradations need to be taken into account, cov...
Minimization of a quadratic form 〈x, Tx〉 + 〈p, x〉 + a under constraints defined by a linear system is a common optimization problem. It is assumed that the operator T is symmetric positive definite or positive semidefinite. Several extensions to different sets of linear matrix constraints are investigated. Solutions of this problem may be given usi...
This paper proposes a method for reconstruction of blurred images damaged by a
separable motion blur. The method can be used after the application of currently developed
image restoration algorithms. Our approach is based on the usage of least squares solutions
of certain matrix equations which define the separable motion blur. The method uses
appr...
We analyze the problem of constrained minimization of the real quadratic functional <x, Tx> +<x, p>+a, subject to the inconsistent system of linear equations Ax = b, where T is a positive definite or positive semidefinite matrix. Both cases are analyzed separately, and respective relationships have been established between the solution of the origi...
We propose an image restoration method. The method generalizes image restoration algorithms that are based on the Moore–Penrose solution of certain matrix equations that define the linear motion blur. Our approach is based on the usage of least squares solutions of these matrix equations, wherein an arbitrary matrix of appropriate dimensions is inc...
Until recently, anti-cancer clinics in order to plan and schedule chemotherapy treatments
applied time-consuming methods, characterized by a high risk of error occurrence. Considering
all the barriers that make chemotherapy treatment and demanding procedure, to evaluate if a
chemotherapy management program could improve the administration of chemot...
In this work a linearly constrained minimization of a positive semidefinite quadratic functional is examined. Our results are concerning infinite dimensional real Hilbert spaces, with a singular positive operator related to the functional, and considering as constraint a singular operator. The difference between the proposed minimization and previo...
In this work we study a minimization problem for a matrix-valued function
under linear constraints, in the case of a singular matrix. The proposed method
differs from others on the restriction of the minimizing matrix to the range of the
corresponding quadratic function. Moreover, we present two applications of the proposed
minimization method in L...
Efficient evaluation of the full-rank QDR decomposition is established. A method and algorithm for efficient symbolic computation of AT,S(2) inverses of a given rational matrix A is defined using the full-rank QDR decomposition of an appropriate rational matrix W. The algorithm is implemented using MATHEMATICA’s ability to deal with symbolic expres...
An efficient algorithm for computing AT,S(2) inverses of a given constant matrix A, based on the QR decomposition of an appropriate matrix W, is presented. Correlations between the derived representation of outer inverses and corresponding general representation based on arbitrary full-rank factorization are derived. In particular cases we derive r...
We introduce the T-restricted weighted generalized inverse of a singular matrix A with respect to a positive semidefinite matrix T, which defines a seminorm for the space. The new approach proposed is that since T is positive semidefinite, the minimal seminorm solution is considered for all vectors perpendicular to the kernel of T.
The aim of this work is to propose a method based on B-splines
for signal filtering and signal reconstruction. The proposed approach
consists of applying the Moore-Penrose inverse for the reconstruction
of noisy signals and the use of smoothing techniques with a Whittaker
smoother to control the roughness of variation in order to extract the signal...
In this article we provide a fast computational method in order to calculate the Moore–
Penrose inverse of singular square matrices and of rectangular matrices. The proposed
method proves to be much faster and has significantly better accuracy than the already
proposed methods, while works for full and sparse matrices.
A common optimization problem is the minimization of a symmetric
positive definite quadratic form hx, Txi under linear constraints. The
solution to this problem may be given using the Moore–Penrose inverse matrix.
In this work at first we extend this result to infinite dimensional complex
Hilbert spaces, where a generalization is given for positive...
EP matrices are a wide class of matrices which, among other things, can be characterized through factorizations. In this paper
we are using two factorization algorithms in order to compute and factorize the Moore-Penrose inverse of a singular EP matrix.
For the implementation of the algorithms we make use of a Computer Algebra System such as Maple....
The field of image restoration has seen a tremendous growth in interest over the last two decades. The recovery of an original image from degraded observations is a crucial method and finds application in several scientific areas including medical imaging and diagnosis, military surveillance, satellite and astronomical imaging, and remote sensing....
In this paper we use the Moore-Penrose inverse in the case of a close to singular and ill-conditioned, or singular variance-covariance matrix, in the classic Portfolio Selection Problem. In this way the possible singularity of the variance-covariance matrix is tackled in an efficient way so that the various application of the Problem to benefit fro...
In this work we use the generalized inverse in the classical Portfolio Selection
Problem,a portfolio consisting of 7 intraday exchange rates which is updated
every minute. In this case, since the variance covariance matrix is very close to
singular and numerically ill-conditioned, the problem is tackled in an efficient
way to benefit from the numer...
This paper presents a fast computational method that finds application
in a broad scientific field such as digital image restoration. The proposed method provides a new approach to the problem of image reconstruction by using the Moore-Penrose inverse. The resolution of the reconstructed image remains at a very high level but the main advantage of...
In this article we present a new computational method for the Moore-Penrose inverse that finds application in a broad scientific field such as digital image restoration. The proposed numerical solution, based upon QR factorization, provide us the Moore-Penrose inverse of any matrix. In fact, this work is a generalization of [7] in the sense that th...
n this article a fast computational method is provided in order to calculate the
Moore-Penrose inverse of full rank m×n matrices and of square matrices with at least one zero row
or column. Sufficient conditions are also given for special type products of square matrices so that
the reverse order law for the Moore-Penrose inverse is satisfied.
In this paper, we characterize EP operators through the existence of different types of factorizations. Our
results extend to EP operators existing characterizations for EP matrices and give new characterizations both
for EP matrices and EP operators
Let H be a complex Hilbert space. In this work we compute the gen-
eralized inverse of a ¯nite rank operator on H and give necessary and su±cient
conditions such that the generalized inverse of the product of two rank-1 oper-
ators is the product of the generalized inverses of the corresponding operators
in reverse order. We also consider the gener...
Let H be a complex Hilbert space and let T : H ! H be a bounded
linear operator with closed range. It is proved that the generalized inverse
T+, of T, is a polynomial of T if and only if T is algebraic and commutes
with T+. It is also given sufficient conditions so that the generalized
inverse T+, belongs to the weakly closed algebra generated by T...
In this work we use the generalized inverse in the classical Portfolio Selection Problem. In this way when the variance covariance matrix is close to singular and numerically ill-conditioned, or singular, the problem is tackled in an efficient way to benefit from the numerical tractability of the Moore-Penrose inverse.
Questions
Question (1)
I need to find out the complexity for the computation of the Moore Penrose inverse of a matrix.
Any ideas?
Projects
Projects (4)
In the last few years, detection has become a powerful methodology for network protection and network security measures. This work will present and explore new detection schemes for data recorded over the network, applicable on the broad scientific field of information security, including intrusion detection and prevention.
The current scheme will be performed and evaluated for the case of the KDD-NSL and the UNSW-NB15 datasets.
EP matrices and operators have the property AA^+ = A^+A, where A+ stands for the Moore- Penrose inverse of a square matrix A. The Study of the properties, factorization, applications and classification of EP matrices are a part of this project.
Computational methods, Symbolically and Numerically. The Moore- Penrose inverse of the Aluthge transform. Properties of the above transforms.
