Two-stage Algorithm for solving Arbitrary Trapezoidal Fully Fuzzy Sylvester Matrix Equations

Many authors proposed analytical methods for solving fully fuzzy Sylvester matrix equation (FFSME) based on Vec-operator and Kronecker product. However, these methods are restricted to nonnegative fuzzy numbers and cannot be extended to FFSME with near-zero fuzzy numbers. The main intention of this paper is to develop a new numerical method for solving FFSME with near-zero trapezoidal fuzzy numbers that provides a wider scope of trapezoidal fully fuzzy Sylvester matrix equation (TrFFSME) in scientific applications. This numerical method can solve the trapezoidal fully fuzzy Sylvester matrix equation with arbitrary coefficients and find all possible finite arbitrary solutions for the system. In order to obtain all possible fuzzy solutions, the TrFFSME is transferred to a system of non-linear equations based on newly developed arithmetic fuzzy multiplication between trapezoidal fuzzy numbers. The fuzzy solutions to the TrFFSME are obtained by developing a new two-stage algorithm. To illustrate the proposed method numerical example is solved.

On the Low-Degree Solution of the Sylvester Matrix Polynomial Equation

We study the low-degree solution of the Sylvester matrix equation A 1 λ + A 0 X λ + Y λ B 1 λ + B 0 = C 0 , where A 1 λ + A 0 and B 1 λ + B 0 are regular. Using the substitution of parameter variables λ , we assume that the matrices A 0 and B 0 are invertible. Thus, we prove that if the equation is solvable, then it has a low-degree solution L λ , M λ , satisfying the degree conditions δ L λ < Ind A 0 − 1 A 1 and δ M λ < Ind B 1 B 0 − 1 .

Arbitrary Coupled Trapezoidal Fully Fuzzy Sylvester Matrix Equation

There are many applications where couple of Sylvester matrix equations (CSME) are required to be solved simultaneously, especially in analyzing the stability of control systems. However, there are some situations in which the crisp CSME are not well equipped to deal with the uncertainty problem during the process of stability analysis in control system engineering. Thus, in this paper a new method for solving a coupled trapezoidal fully fuzzy Sylvester matrix equation (CTrFFSME) with arbitrary coefficients is proposed. The arithmetic fuzzy multiplication operation is applied to convert the CTrFFSME into a system of non-linear equations. Then the obtained non-linear system is reduced and converted to a system of absolute equations where the fuzzy solution is obtained by solving that system. The proposed method can solve many unrestricted fuzzy systems such as Sylvester and Lyapunov fully fuzzy matrix equations with triangular and trapezoidal fuzzy numbers. We illustrate the proposed methods by solving numerical example.

Trapezoidal Fully Fuzzy Sylvester Matrix Equation with Arbitrary Coefficients

Almost every existing method for solving trapezoidal fully fuzzy Sylvester matrix equation restricts the coefficient matrix and the solution to be positive fuzzy numbers only. In this paper, we develop new analytical methods to solve a trapezoidal fully fuzzy Sylvester matrix equation with restricted and unrestricted coefficients. The trapezoidal fully fuzzy Sylvester matrix equation is transferred to a system of crisp equations based on the sign of the coefficients by using Ahmd arithmetic multiplication operations between trapezoidal fuzzy numbers. The constructed method not only obtain a simple crisp system of linear equation that can be solved by any classical methods but also provide a widen the scope of the trapezoidal fully fuzzy Sylvester matrix equation in scientific applications. Furthermore, these methods have less steps and conceptually easy to understand when compared with existing methods. To illustrate the proposed methods numerical examples are solved.

Three special kinds of least squares solutions for the quaternion generalized Sylvester matrix equation

<abstract><p>In this paper, we propose an efficient method for some special solutions of the quaternion matrix equation $ AXB+CYD = E $. By integrating real representation of a quaternion matrix with $ \mathcal{H} $-representation, we investigate the minimal norm least squares solution of the previous quaternion matrix equation over different constrained matrices and obtain their expressions. In this way, we first apply $ \mathcal{H} $-representation to solve quaternion matrix equation with special structure, which not only broadens the application scope of $ \mathcal{H} $-representation, but further expands the research idea of solving quaternion matrix equation. The algorithms only include real operations. Consequently, it is very simple and convenient, and it can be applied to all kinds of quaternion matrix equation with similar problems. The numerical example is provided to illustrate the feasibility of our algorithms.</p></abstract>