Kronecker Map Research Articles

Solutions to the linear transpose matrix equations and their application in control

In this paper, we study the solutions to the linear transpose matrix equations $$AX+X^{T}B=C$$ and $$AX+X^{T}B=CY$$ , which have many important applications in control theory. By applying Kronecker map and Sylvester sum, we obtain some necessary and sufficient conditions for existence of solutions and the expressions of explicit solutions for the Sylvester transpose matrix equation $$AX+X^{T}B=C$$ . Our conditions only need to check the eigenvalue of $$ B^{T}A^{-1}$$ , and, therefore, are simpler than those reported in the paper (Piao et al. in J Frankl Inst 344:1056–1062, 2007). The corresponding algorithms permit the coefficient matrix C to be any real matrix and remove the limit of $$C=C^{T}$$ in Piao et al. Moreover, we present the solvability and the expressions of parametric solutions for the generalized Sylvester transpose matrix equation $$AX+X^{T}B=CY$$ using an alternative approach. A numerical example is given to demonstrate that the introduced algorithm is much faster than the existing method in the paper (De Teran and Dopico in 434:44–67;2011). Finally, the continuous zeroing dynamics design of time-varying linear system is provided to show the effectiveness of our algorithm in control.

A combination of Sylvester block sum and block matrix Kronecker map for explicit solutions of Sylvester system of matrix equations

In this paper, we consider an explicit solution of system of Sylvester matrix equations of the form A1V1 − E1V1F1 = B1W1 and A2V2 − E2V2F2 = B2W2 with F1 and F2 being arbitrary matrices, where V1,W1,V2 and W2 are the matrices to be determined. First, the definitions, of the matrix polynomial of block matrix, Sylvester sum, and Kronecker product of block matrices are defined. Some definitions, lemmas, and theorems that are needed to propose our method are stated and proved. Numerical test problems are solved to illustrate the suggested technique.

Explicit and Iterative Methods for Solving the Matrix Equation AV + BW = EVF + C

AbstractIn this paper, we consider explicit and iterative methods for solving the Generalized Sylvester matrix equation AV + BW = EVF + C. Based on the use of Kronecker map and Sylvester sum some lemmas and theorems are stated and proved where explicit and iterative solutions are obtained. The proposed methods are illustrated by numerical example. The obtained results show that the methods are very neat and efficient.

On solutions of the generalized Stein quaternion matrix equation

In this paper, some solution expressions to the quaternion matrix equation X−AXF=BY are obtained. They are the further conclusions of the paper (Song et al. in Int. J. Comput. Math. 89:890–900, 2012). By applying of Kronecker map and complex representation of a quaternion matrix, the sufficient conditions for finding the solution have been established. At the end of the article, numerical examples show the applications of the proposed explicit solution.

On solutions of quaternion matrix equations XF #AX = BY and XF #A = BY

In this paper, the quaternion matrix equations XF #AX = BY and XF #A▪ = BY are investigated. For convenience, they were called generalized Sylvesterquaternion matrix equation and generalized Sylvester-j-conjugate quaternion matrix equation, which include the Sylvester matrix equation and Lyapunov matrix equation as special cases. By applying of Kronecker map and complex representation of a quaternion matrix, the sufficient conditions to compute the solution can be given and the expressions of the explicit solutions to the above two quaternion matrix equations XF #AX = BY and XF #A▪ = BY are also obtained. By the established expressions, it is easy to compute the solution of the quaternion matrix equation in the above two forms. In addition, two practical algorithms for these two quaternion matrix equations are give. One is complex representation matrix method and the other is a direct algorithm by the given expression. Furthermore, two illustrative examples are proposed to show the efficiency of the given method.

Explicit solutions to the quaternion matrix equations X−AXF=C and X−A[Xtilde] F=C

In the present paper, we investigate the quaternion matrix equation X−AXF=C and X−A[Xtilde] F=C. For convenience, we named the quaternion matrix equations X−AXF=C and X−A[Xtilde] F=C as quaternion Stein matrix equation and quaternion Stein-conjugate matrix equation. Based on the Kronecker map and complex representation of a quaternion matrix, we give the solution expressions of the quaternion Stein matrix equation and quaternion Stein-conjugate matrix equation. Through these expressions, we can easily obtain the solution of the above two equations. In order to compare the direct algorithm with the indirect algorithm, we propose an example to illustrate the effectiveness of the proposed method.

On solutions of matrix equation XF−AX=C and $XF-A\widetilde{X}=C$ over quaternion field

By means of Kronecker map and complex representation of a quaternion matrix, some explicit solutions to the quaternion matrix equations XF−AX=C and \(XF-A\widetilde{X}=C\) are established. One of the solutions is neatly expressed by a symmetric matrix, a controllability matrix and an observability matrix. In addition, two practical algorithms for these two equations are given.

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 − A V F = B W and Sylvester matrix equations A V − E V F = B W , A V + B W = E V F and A V − V F = B W . 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.

On matrix equations [formula omitted] and [formula omitted

With the help of the concept of Kronecker map, an explicit solution for the matrix equation X − A X F = C is established. This solution is neatly expressed by a symmetric operator matrix, a controllability matrix and an observability matrix. In addition, the matrix equation X − A X ¯ F = C is also studied. An explicit solution for this matrix equation is also proposed by means of the real representation of a complex matrix. This solution is neatly expressed by a symmetric operator matrix, two controllability matrices and two observability matrices.

Solving the generalized Sylvester matrix equation AV + BW = V F via Kronecker map

This note considers the solution to the generalized Sylvester matrix equation AV + BW = V F with F being an arbitrary matrix, where V and W are the matrices to be determined. With the help of the Kronecker map, an explicit parametric solution to this matrix equation is established. The proposed solution possesses a very simple and neat form, and allows the matrix F to be undetermined.

Solving the generalized Sylvester matrix equation [formula omitted] via a Kronecker map

This note considers the solution to the generalized Sylvester matrix equation A V + B W = E V F with F being an arbitrary matrix, where V and W are the matrices to be determined. With the help of the Kronecker map, some properties of the Sylvester sum are first proposed. By applying the Sylvester sum as tools, an explicit parametric solution to this matrix equation is established. The proposed solution is expressed by the Sylvester sum, and allows the matrix F to be undetermined.

On solutions of matrix equations [formula omitted] and [formula omitted

With the help of the Kronecker map, a complete, general and explicit solution to the Yakubovich matrix equation V − A V F = B W , with F in an arbitrary form, is proposed. The solution is neatly expressed by the controllability matrix of the matrix pair ( A , B ) , a symmetric operator matrix and an observability matrix. Some equivalent forms of this solution are also presented. Based on these results, explicit solutions to the so-called Kalman–Yakubovich equation and Stein equation are also established. In addition, based on the proposed solution of the Yakubovich matrix equation, a complete, general and explicit solution to the so-called Yakubovich-conjugate matrix is also established by means of real representation. Several equivalent forms are also provided. One of these solutions is neatly expressed by two controllability matrices, two observability matrices and a symmetric operator matrix.

On solutions of the matrix equations XF − AX = C and [formula omitted

With the help of the concept of Kronecker map, an explicit solution for the matrix equation XF − AX = C is established. This solution is neatly expressed by a symmetric operator matrix, a controllability matrix and an observability matrix. In addition, the matrix equation XF - A X ¯ = C is also studied. An explicit solution for this equation is also proposed by means of the real representation of a complex matrix. This solution is neatly expressed by a symmetric operator matrix, two controllability matrices and two observability matrices.

Quantum Unique Ergodicity for Maps on the Torus

When a map is classically uniquely ergodic, it is expected that its quantization will posses quantum unique ergodicity. In this paper we give examples of Quantum Unique Ergodicity for the perturbed Kronecker map, and an upper bound for the rate of convergence.

Simple Multivariate Polynomial Multiplication

We observe that polynomial evaluation and interpolation can be performed fast over a multidimensional grid (lattice), and we apply this observation in order to devise a simple algorithm for multivariate polynomial multiplication. Surprisingly, this simple idea enables us to improve the known algorithms for multivariate polynomial multiplication based on the forward and backward application of Kronecker's map; in particular, we decrease, by the factor log log N, the known upper bound on the arithmetic time-complexity of this computation (over any field of constants), provided that the degree d in each of the m variables is fixed, m grows to the infinity, and N = ( d + 1) m .