Real Matrix Equations Research Articles

An Efficient Method for Split Quaternion Matrix Equation X − Af(X)B = C

In this paper, we consider the split quaternion matrix equation X−Af(X)B=C, f(X)∈{X,XH,XiH,XjHXkH}. The H representation method has the characteristics of transforming a matrix with a special structure into a column vector with independent elements. By using the real representation of split quaternion matrices, H representation method, the Kronecker product of matrices and the Moore-Penrose generalized inverse, we convert the split quaternion matrix equation into the real matrix equation, and derive the sufficient and necessary conditions and the general solution expressions for the (skew) bisymmetric solution of the original equation. Moreover, we provide numerical algorithms and illustrate the efficiency of our method by two numerical examples.

The semi-tensor product method for special least squares solutions of the complex generalized Sylvester matrix equation

<abstract><p>In this paper, we are interested in the minimal norm of least squares Hermitian solution and the minimal norm of least squares anti-Hermitian solution for the complex generalized Sylvester matrix equation $ CXD+EXF = G $. By utilizing of the real vector representations of complex matrices and the semi-tensor product of matrices, we first transform solving special least squares solutions of the above matrix equation into solving the general least squares solutions of the corresponding real matrix equations, and then obtain the expressions of the minimal norm of least squares Hermitian solution and the minimal norm of least squares anti-Hermitian solution. Further, we give two numerical algorithms and two numerical examples, and numerical examples illustrate that our proposed algorithms are more efficient and accurate.</p></abstract>

An efficient real representation method for least squares problem of the quaternion constrained matrix equation AXB + CY D = E

Let and represent the sets of all η-Hermitian quaternion matrices and η-anti-Hermitian quaternion matrices, respectively. On the basis of the real representation matrix of a quaternion matrix and its particular structure, we convert the least squares problem of the quaternion matrix equation AXB + CY D = E over into the corresponding problem of the real matrix equation over free variables, and then we establish its unique minimal norm least squares solution. Our resulting expressions are expressed only by real matrices, and the algorithm only includes real operations. Consequently, they are very simple and convenient. Compared with the existing method [S.F. Yuan, Q.W. Wang, and X. Zhang, Least-squares problem for the quaternion matrix equation AXB + CYD = E over different constrained matrices, Int. J. Comput. Math. 90 (2013), pp. 565–576], the final two examples show that our method is more efficient and superior.

An efficient method for least-squares problem of the quaternion matrix equation X - AX̂B = C

ABSTRACT In this paper, we consider the quaternion matrix equation , and study its minimal norm least squares solution, j-self-conjugate least-squares solution and anti-j-self-conjugate least-squares solution. By the real representation matrices of quaternion matrices, their particular structure and the properties of Frobenius norm, we convert above least-squares problems into corresponding problems of real matrix equations. The final results of the expressions only involve real matrices, and thus, the corresponding algorithms only involve real operations. Compared with the existing results, they are more convenient and efficient, which are also illustrated by the last two numerical examples.

On the split quaternion matrix equation $$AX=B$$

In the paper, we consider the split quaternion matrix equation $$AX=B$$. We design several real representations of split quaternion matrix to transform the above split quaternion matrix equation into some real matrix equations. By using this method, we give some necessary and sufficient conditions for $$AX=B$$ to have a X or $$X=\pm X^{\star }$$ solution and derive the expressions of solutions when equation is solvable, where $$X^{\star } \in \{X^*, X^\eta , X^{\eta *}\}$$, $$X^*$$ is the conjugate transpose of X, for $$\eta \in \{i, j, k\}$$, $$ X^{\eta }, X^{\eta *}$$ are $$\eta $$-conjugate, $$\eta $$-Hermitian of X, respectively. We also present the solvability conditions and expression of the unique solution X or $$X=\pm X^{\star }$$.

Use of dual numbers in kinematical analysis of spatial mechanisms. Part II: applying the method for the generalised Cardan mechanism

The equations which permit obtaining the displacements from kinematical pairs were deduced in the first part of the paper. An actual case is presented in the second part of the work, the method being applied for the generalized Cardan mechanism. In a first stage, the parameters required for the completion of the dual matrices from the dual matrix equation of closing the kinematical chain are identified. Next, the matrix equation which allow for finding the rotations and the translations, respectively, from the pairs, are identified. A special attention is assumed on establishing the rotations from the pairs. The dual matrix equation is split into two real matrix equations, corresponding to rotations and to the translational motions, respectively. The equation corresponding to the translations presents a complex form, but it is simplified by using some characteristics of the matrices. Additionally, this matrix equation is written under a form which allows for algorithmization. Finally, there are obtained the equations which describe the position of the mechanism - the same as in the classical approach.

Consistency of Split Quaternion Matrix Equations $$AX^{\star }-XB=CY+D$$ and $$X-AX^\star B=CY+D$$

In this paper, the split quaternion matrix equations $$AX^{\star }-XB=CY+D$$ and $$X-AX^{\star }B=CY+D$$ are considered, where $$X^\star $$ is X or the $$\phi $$ -conjugate of X, $$\phi \in \{i, j, k\}$$ . We design some new real representations of a split quaternion matrix, which enable us to convert $$\phi $$ -conjugate matrix equations into some real matrix equations. By using this method, we show that the original split quaternion matrix equation is consistent if and only if its corresponding real matrix equation is consistent. Moreover, we formulate the split quaternion solutions by the real solutions of its corresponding real matrix equation. As applications, we derive the necessary and sufficient conditions for the existence of solutions to the above two split quaternion matrix equations and their special cases $$AX^{\star }-XB=C$$ and $$X-AX^{\star }B=C$$ . We also obtain some conditions for the mentioned four split matrix equations to be uniquely solvable.

Consistency of Quaternion Matrix Equations $AX^{\star}-XB=C$ and $X-AX^\star B=C$

For a given ordered units triple $\{q_1, q_2, q_3\}$, the solutions to the quaternion matrix equations $AX^{\star}-XB=C$ and $X-AX^{\star}B=C$, $X^{\star} \in \{ X , X^{\eta} , X^* , X^{\eta*}\}$, where $X^*$ is the conjugate transpose of $X$, $X^{\eta}=-\eta X \eta$ and $X^{\eta*}=-\eta X^* \eta$, $\eta \in \{q_1, q_2, q_3\}$, are discussed. Some new real representations of quaternion matrices are used, which enable one to convert $\eta$-conjugate (transpose) matrix equations into some real matrix equations. By using this idea, conditions for the existence and uniqueness of solutions to the above quaternion matrix equations are derived. Also, methods to construct the solutions from some related real matrix equations are presented.

On solutions to the matrix equations [formula omitted] and [formula omitted

The solution of the generalized Sylvester real matrix equation XB−AX=CY is important in stability analysis and controller design in linear systems. This paper presents an explicit solution to the generalized Sylvester real matrix equation XB−AX=CY. Based on the derived explicit solution to the considered generalized Sylvester real matrix equation, a new approach is provided for obtaining the solutions to the generalized Sylvester quaternion j-conjugate matrix equation XB−AX^=CY using the real representation of a quaternion matrix. The closed form solution is established in an explicit form for this generalized Sylvester quaternion j-conjugate matrix equation. The existing complex representation method requires the coefficient matrix A to be a block diagonal matrix over complex field, while it is any suitable dimension quaternion matrix in the present paper. Therefore, we generalize the existing results.

Explicit solutions to the matrix equation E X F − AX = C

The matrix equation E X F − AX = C is investigated in this study, and three approaches are provided to solve this equation. The first approach is to transform it into a real matrix equation with the help of real representation of complex matrices. In the second approach, the solution is given in terms of characteristic polynomial of a constructed matrix pair. In the third approach, the solution can be neatly expressed in terms of controllability matrices and observability matrices. By specialising the obtained solutions of E X F − AX = C, some new expressions of the generalised Sylvester matrix equations are also provided.

Real-domain decoder for full-rate full-diversity STBC with multidimensional constellations

In this letter, we present a new maximum likelihood (ML) decoding algorithm for space time block codes (STBCs) that employ multidimensional constellations. We start with a lattice representation for STBCs which transforms complex channel models into real matrix equations. Based on the lattice representation, we propose a new decoding algorithm for quasiorthogonal STBCs (QO-STBC) which allows simpleML decoding with performance identical to the conventional ML decoder. Multidimensional rotated constellations are constructed for the QO-STBCs to achieve full diversity. As a consequence, for quasi-orthogonal designs with an arbitrary number of transmit antennas N (N ≤ 4), the proposed decoding scheme achieves full rate and full diversity while reducing the decoding complexity from 𝒪(M <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">c</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N/2</sup> ) to 𝒪(M <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">c</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N/4</sup> ) in a M <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">c</sub> -QAM constellation.

Calculation Methods for Current Distribution in Single-Turn Coils of Arbitrary Cross Section

Three methods to calculate the current distribution, both in magnitude and phase, throughout the cross section of a single-turn coil are presented. The methods are analytical for computer computations in that exact solutions would be obtained if round-off errors could be avoided completely. The numerical procedure, which solves the resulting matrix system equation, common to all three solutions, is discussed. It provides a numerically stable solution by separating the complex matrix equation into two real matrix equations, namely, the real and imaginary parts.