Ahmed M. E. Bayoumi

On the explicit solutions of forms of the Sylvester and the Yakubovich matrix equations

In this paper, we consider the explicit solutions of two matrix equations, namely, the Yakubovich matrix equation V-AVF=BW and Sylvester matrix equations AV-EVF=BW,AV+BW=EVF and AV-VF=BW. For this purpose, we make use of Kronecker map and Sylvester sum as well as the concept of coefficients of characteristic polynomial of the matrix A. Some lemmas and theorems are stated and proved where explicit and parametric solutions are obtained. The proposed methods are illustrated by numerical examples. The results obtained show that the methods are very neat and efficient.

Two iterative algorithms for the reflexive and Hermitian reflexive solutions of the generalized Sylvester matrix equation

This paper is concerned with iterative solutions to the generalized Sylvester matrix equation A 1 V + A 2 V ¯ + B 1 W + B 2 W ¯ = E 1 VF 1 + E 2 V ¯ F 2 + C . Two iterative algorithms are presented to obtain the reflexive and Hermitian reflexive solutions. With these iterative algorithms, for any initial reflexive and Hermitian reflexive matrices the solutions can be obtained. Some needed lemmas are first stated, then two theorems are stated and proved where the iterative solutions are obtained. Finally, we report two numerical examples to verify the theoretical results.

A finite iterative algorithm for the solution of Sylvester-conjugate matrix equations AV+BW=EV¯F+C and AV+BW¯=EV¯F+C

In this paper, we consider two iterative algorithms for the Sylvester-conjugate matrix equation AV+BW=EV¯F+C and AV+BW¯=EV¯F+C. When these two matrix equations are consistent, for any initial matrices the solutions can be obtained within finite iterative steps in the absence of round off errors. Some lemmas and theorems are stated and proved where the iterative solutions are obtained. Two numerical examples are given to illustrate the effectiveness of the proposed method.

Finite Iterative Algorithm for Solving a Complex of Conjugate and Transpose Matrix Equation

We consider an iterative algorithm for solving a complex matrix equation with conjugate and transpose of two unknowns of the form:

Finite iterative Hamiltonian solutions of the generalized coupled Sylvester – conjugate matrix equations

In this paper, we present an iterative algorithm to solve a generalized coupled Sylvester – conjugate matrix equations over Hamiltonian matrices. When the considered systems of matrix equations are consistent, it is proven that the solution can be obtained within finite iterative steps for any arbitrary initial generalized Hamiltonian matrices in the absence of round off errors. Two numerical examples are given to illustrate the effectiveness of the proposed method.

An accelerated gradient-based iterative algorithm for solving extended Sylvester–conjugate matrix equations

In this paper, we present an accelerated gradient-based iterative algorithm for solving extended Sylvester–conjugate matrix equations. The idea is from the gradient-based method introduced in Wu et al. (Applied Mathematics and Computation 217(1): 130–142, 2010a) and the relaxed gradient-based algorithm proposed in Ramadan et al. (Asian Journal of Control 16(5): 1–8, 2014) and the modified gradient-based algorithm proposed in Bayoumi (PhD thesis, Ain Shams University, 2014). The convergence analysis of the algorithm is investigated. We show that the iterative solution converges to the exact solution for any initial value provided some appropriate assumptions be made. A numerical example is given to illustrate the effectiveness of the proposed method and to test its efficiency and accuracy compared with an existing one presented in Wu et al. (2010a), Ramadan et al. (2014) and Bayoumi (2014).

Finite iterative Hermitian R-conjugate solutions of the generalized coupled Sylvester-conjugate matrix equations

Abstract In this paper, an iterative algorithm for solving a generalized coupled Sylvester-conjugate matrix equations over Hermitian R -conjugate matrices given by A 1 V B 1 + C 1 W D 1 = E 1 V ¯ F 1 + G 1 and A 2 V B 2 + C 2 W D 2 = E 2 V ¯ F 2 + G 2 is presented. When these two matrix equations are consistent, the convergence theorem shows that a solution can be obtained within finite iterative steps in the absence of round-off error for any initial arbitrary Hermitian R -conjugate solution matrices V 1 , W 1 . Some lemmas and theorems are stated and proved where the iterative solutions are obtained. A numerical example is given to demonstrate the behavior of the proposed method and to support the theoretical results.