Leila Lebtahi

The diamond partial order in rings

Abstract In this paper we introduce a new partial order on a ring, namely the diamond partial order. This order is an extension of a partial order defined in a matrix setting in [J.K. Baksalary and J. Hauke, A further algebraic version of Cochran’s theorem and matrix partial orderings, Linear Algebra and its Applications, 127, 157–169, 1990]. We characterize the diamond partial order on rings and study its relationships with other partial orders known in the literature. We also analyse successors, predecessors and maximal elements under the diamond order.

The inverse eigenvalue problem for a Hermitian reflexive matrix and the optimization problem

The inverse eigenvalue problem and the associated optimal approximation problem for Hermitian reflexive matrices with respect to a normal { k + 1 } -potent matrix are considered. First, we study the existence of the solutions of the associated inverse eigenvalue problem and present an explicit form for them. Then, when such a solution exists, an expression for the solution to the corresponding optimal approximation problem is obtained.

Algorithms for {K,s+1}-potent matrix constructions

In this paper, we deal with {K,s+1}-potent matrices. These matrices generalize all the following classes of matrices: k-potent matrices, periodic matrices, idempotent matrices, involutory matrices, centrosymmetric matrices, mirrorsymmetric matrices, circulant matrices, among others. Several applications of these classes of matrices can be found in the literature. We develop algorithms in order to compute {K,s+1}-potent matrices and {K,s+1}-potent linear combinations of {K,s+1}-potent matrices. In addition, some examples are presented in order to show the numerical performance of the method.

Inverse eigenvalue problem for normal J-hamiltonian matrices

Abstract A complex square matrix A is called J -hamiltonian if A J is hermitian where J is a normal real matrix such that J 2 = − I n . In this paper we solve the problem of finding J -hamiltonian normal solutions for the inverse eigenvalue problem.

Special elements in a ring related to Drazin inverses

In this article, the existence of the Drazin (group) inverse of an element a in a ring is analysed when a m k = ka n , for some unit k and m, n ∈ ℕ. The same problem is studied for the case when a* = ka m k −1 and for the {k, s + 1}-potent elements. In addition, relationships with other special elements of the ring are also obtained.

Generalized centro-invertible matrices with applications

Abstract Centro-invertible matrices are introduced by R.S. Wikramaratna in 2008. For an involutory matrix R , we define the generalized centro-invertible matrices with respect to R to be those matrices A such that R A R = A − 1 . We apply these matrices to a problem in modular arithmetic. Specifically, algorithms for image blurring/deblurring are designed by means of generalized centro-invertible matrices. In addition, if R 1 and R 2 are n × n involutory matrices, then there is a simple bijection between the set of all centro-invertible matrices with respect to R 1 and the set with respect to R 2 .

Properties of a matrix group associated to a {K,s+1}-potent matrix

In a previous paper, the authors introduced and characterized a new kind of matrices called {K,s+1}-potent. In this paper, an associated group to a {K, s+1}-potent matrix is explicitly constructed and its properties are studied. Moreover, it is shown that the group is a semidirect product of Z2 acting on Z(s+1)2−1. For some values of s, more specifications on the group are derived. In addition, some illustrative examples are given.