Reflexive Solutions Research Articles

Finite iterative algorithms for solving generalized coupled Sylvester systems – Part I: One-sided and generalized coupled Sylvester matrix equations over generalized reflexive solutions

The generalized coupled Sylvester systems play a fundamental role in wide applications in several areas, such as stability theory, control theory, perturbation analysis, and some other fields of pure and applied mathematics. The iterative method is an important way to solve the generalized coupled Sylvester systems. In this two-part article, finite iterative methods are proposed for solving one-sided (or two-sided) and generalized coupled Sylvester matrix equations and the corresponding optimal approximation problem over generalized reflexive solutions (or reflexive solutions). In part I, an iterative algorithm is constructed to solve one-sided and coupled Sylvester matrix equations ( AY − ZB, CY − ZD) = ( E, F) over generalized reflexive matrices Y and Z. When the matrix equations are consistent, for any initial generalized reflexive matrix pair [ Y 1, Z 1], the generalized reflexive solutions can be obtained by the iterative algorithm within finite iterative steps in the absence of round-off errors, and the least Frobenius norm generalized reflexive solution pair can be obtained by choosing a special kind of initial matrix pair. The unique optimal approximation generalized reflexive solution pair [ Y ^ , Z ^ ] to a given matrix pair [ Y 0, Z 0] in Frobenius norm can be derived by finding the least-norm generalized reflexive solution pair [ Y ∼ ∗ , Z ∼ ∗ ] of two new corresponding generalized coupled Sylvester matrix equations ( A Y ∼ - Z ∼ B , C Y ∼ - Z ∼ D ) = ( E ∼ , F ∼ ) , where E ∼ = E - AY 0 + Z 0 B , F ∼ = F - CY 0 + Z 0 D . Several numerical examples are given to show the effectiveness of the presented iterative algorithm.

A finite iterative algorithm for solving the generalized [formula omitted]-reflexive solution of the linear systems of matrix equations

In this paper, we proposed an algorithm for solving the linear systems of matrix equations { ∑ i = 1 N A i ( 1 ) X i B i ( 1 ) = C ( 1 ) , ⋮ ∑ i = 1 N A i ( M ) X i B i ( M ) = C ( M ) . over the generalized ( P , Q ) -reflexive matrix X l ∈ R n × m ( A l ( i ) ∈ R p × n , B l ( i ) ∈ R m × q , C ( i ) ∈ R p × q , l = 1 , 2 , … , N , i = 1 , 2 , … , M ). According to the algorithm, the solvability of the problem can be determined automatically. When the problem is consistent over the generalized ( P , Q ) -reflexive matrix X l ( l = 1 , … , N ) , for any generalized ( P , Q ) -reflexive initial iterative matrices X l ( 0 ) ( l = 1 , … , N ) , the generalized ( P , Q ) -reflexive solution can be obtained within finite iterative steps in the absence of roundoff errors. The unique least-norm generalized ( P , Q ) -reflexive solution can also be derived when the appropriate initial iterative matrices are chosen. A sufficient and necessary condition for which the linear systems of matrix equations is inconsistent is given. Furthermore, the optimal approximate solution for a group of given matrices Y l ( l = 1 , … , N ) can be derived by finding the least-norm generalized ( P , Q ) -reflexive solution of a new corresponding linear system of matrix equations. Finally, we present a numerical example to verify the theoretical results of this paper.

The reflexive solution to the system of Sylvester-conjugate matrix equations

The reflexive solution to the system of Sylvester-conjugate matrix equations

Two algorithms for finding the Hermitian reflexive and skew-Hermitian solutions of Sylvester matrix equations

In this paper, we propose two iterative algorithms for finding the Hermitian reflexive and skew-Hermitian solutions of the Sylvester matrix equation A X + X B = C , respectively. We prove that the first (second) algorithm converges to the Hermitian reflexive (skew-Hermitian) solution for any initial Hermitian reflexive (skew-Hermitian) matrix. Finally, two numerical examples illustrate the theoretical results.

On the reflexive and anti-reflexive solutions of the generalised coupled Sylvester matrix equations

The generalised coupled Sylvester matrix equations with unknown matrices X and Y, have important applications in control and system theory. Also it is well known that the reflexive and anti-reflexive matrices have wide applications in many fields. In this article, we consider the generalised coupled Sylvester matrix equations over reflexive and anti-reflexive matrices. First we propose two new matrix equations equivalent to the generalised coupled Sylvester matrix equations over reflexive and anti-reflexive matrices, respectively. Then two new iterative algorithms are proposed for solving these matrix equations. The convergence analysis of the proposed iterative algorithms is derived. Finally, some numerical examples are presented to illustrate the theoretical results of this article.

ON THE REFLEXIVE SOLUTIONS OF THE MATRIX EQUATION AXB + CYD = E

A matrix <TEX>$P{\in}\mathbb{C}^{n{\times}n}$</TEX> is called a generalized reflection matrix if <TEX>$P^*$</TEX> = P and <TEX>$P^2$</TEX> = I. An <TEX>$n{\times}n$</TEX> complex matrix A is said to be a reflexive (anti-reflexive) matrix with respect to the generalized reflection matrix P if A = PAP (A = -PAP). It is well-known that the reflexive and anti-reflexive matrices with respect to the generalized reflection matrix P have many special properties and widely used in engineering and scientific computations. In this paper, we give new necessary and sufficient conditions for the existence of the reflexive (anti-reflexive) solutions to the linear matrix equation AXB + CY D = E and derive representation of the general reflexive (anti-reflexive) solutions to this matrix equation. By using the obtained results, we investigate the reflexive (anti-reflexive) solutions of some special cases of this matrix equation.

Finite iterative algorithms for the reflexive and anti-reflexive solutions of the matrix equation [formula omitted

A matrix P ∈ R n × n is called a generalized reflection matrix if P T = P and P 2 = I . An n × n matrix A is said to be a reflexive (anti-reflexive) matrix with respect to the generalized reflection matrix P if A = P A P ( A = − P A P ) . In this paper, three iterative algorithms are proposed for solving the linear matrix equation A 1 X 1 B 1 + A 2 X 2 B 2 = C over reflexive (anti-reflexive) matrices X 1 and X 2 . When this matrix equation is consistent over reflexive (anti-reflexive) matrices, for any reflexive (anti-reflexive) initial iterative matrices, the reflexive (anti-reflexive) solutions can be obtained within finite iterative steps in the absence of roundoff errors. By the proposed iterative algorithms, the least Frobenius norm reflexive (anti-reflexive) solutions can be derived when spacial initial reflexive (anti-reflexive) matrices are chosen. Furthermore, we also obtain the optimal approximation reflexive (anti-reflexive) solutions to the given reflexive (anti-reflexive) matrices in the solution set of the matrix equation. Finally, some numerical examples are presented to support the theoretical results of this paper.

The Hermitian reflexive solutions to the matrix inverse problem [formula omitted

In this paper, the necessary and sufficient conditions for the solvability of the matrix inverse problem AX = B in Hermitian reflexive matrix set HJ n × n are given, and the solution expression of this matrix inverse problem is also presented. Moreover, an optimal approximation to any matrix in the above solution set in Frobenius norm is provided.

An iterative algorithm for the reflexive solutions of the generalized coupled Sylvester matrix equations and its optimal approximation

The generalized coupled Sylvester matrix equations ( AY - ZB , CY - ZD ) = ( E , F ) with unknown matrices Y , Z are encountered in many systems and control applications. Also these matrix equations have several applications relating to the problem of computing stable eigendecompositions of matrix pencils. In this work, we construct an iterative algorithm to solve the generalized coupled Sylvester matrix equations over reflexive matrices Y , Z . And when the matrix equations are consistent, for any initial matrix pair [ Y 0 , Z 0 ] , a reflexive solution pair can be obtained within finite iteration steps in the absence of roundoff errors, and the least Frobenius norm reflexive solution pair can be obtained by choosing a special kind of initial matrix pair. Also we obtain the optimal approximation reflexive solution pair to a given matrix pair [ Y ¯ , Z ¯ ] in the reflexive solution pair set of the generalized coupled Sylvester matrix equations ( AY - ZB , CY - ZD ) = ( E , F ) . Moreover, several numerical examples are given to show the efficiency of the presented iterative algorithm.

The Generalized Reflexive Solution for a Class of Matrix Equations ( AX=B, XC=D)

In this article, the generalized reflexive solution of matrix equations ( AX=B, XC=D) is considered. With special properties of generalized reflexive matrices, the necessary and sufficient conditions for the solvability and the general expression of the solution are obtained. Moreover, the related optimal approximation problem to a given matrix over the solution set is solved.

Reflexive solution to a system of matrix equations

We derive necessary and sufficient conditions for the existence and an expression of the (anti)reflexive solution with respect to the nontrivial generalized reflection matrix P to the system of complex matrix equations AX = B and XC = D. The explicit solutions of the approximation problem \(\mathop {\min }\limits_{X \in \phi } \) ‖X − E‖F was given, where E is a given complex matrix and ϕ is the set of all reflexive (or antireflexive) solutions of the system mentioned above, and ‖·‖ is the Frobenius norm. Furthermore, it was pointed that some results in a recent paper are special cases of this paper.

An efficient algorithm for the least-squares reflexive solution of the matrix equation A1XB1 = C1, A2XB2 = C2

A n × n real matrix P is said to be a real generalized reflection matrix if P T = P and P 2 = I. A n × n real matrix X is said to be a reflexive matrix with respect to the generalized reflection matrix P if X = PXP. In this paper, the concept of gradient matrix (∇ F( X)) is presented and an algorithm is constructed to solve the reflexive with respect to the generalized reflection matrix P solution of the minimum Frobenius norm residual problem: A 1 XB 1 A 2 XB 2 - C 1 C 2 = min . By this algorithm, for any initial reflexive matrix X 1, a solution X ∗ can be obtained within finite iteration steps in the absence of roundoff errors, and the solution X ∗ with least-norm can be obtained by choosing a special kind of initial reflexive matrix. In addition, in the solution set of above problem, the unique optimal approximation solution X ^ to a given matrix X 0 in Frobenius norm can be obtained by finding the least-norm reflexive solution X ∼ ∗ of the new minimum residual problem: min A 1 X ∼ B 1 A 2 X ∼ B 2 - C 1 ∼ C 2 ∼ , where C 1 ∼ = C 1 - A 1 X 0 B 1 , C 2 ∼ = C 2 - A 2 X 0 B 2 . Given numerical examples show that the iterative method is quite efficient.

The reflexive solutions of the matrix equation AX B = C

In this paper, we study the existence of a reflexive, with respect to the generalized reflection matrix P, solution of the matrix equation AX B = C. For the special case when B = I, we get the result of Peng and Hu [1].

Classroom Note:Centrosymmetric Matrices

Attention is drawn to some earlier work on the odd--even properties of eigenvectors of centrosymmetric matrices.

Substantive and Reflexive Elements in Modern Law

The most comprehensive efforts to develop a new evolutionary approach to law are found in the work of Nonet and Selznick in the United States and Habermas and Luhmann in Germany. While these theorists are concerned with a common problem—the crisis of formal rationality of law—they differ drastically in their accounts of the problem and their vision of the future. This paper tries to resolve these differences by first decomposing and then restructuring the diverse neo-evolutionary models. Using a more comprehensive model of socio-legal covariation, the author identifies an emerging kind of legal structure which he calls “reflexive law.” Reflexive law is characterized by a new kind of legal self-restraint. Instead of taking over regulatory responsibility for the outcome of social processes, reflexive law restricts itself to the installation, correction, and redefinition of democratic self-regulatory mechanisms. The author identifies areas of private law in which reflexive solutions are arguably emerging, and he spells out the consequences which a concern for reflexivity has for a renewed sociological jurisprudence.