mth roots of H-selfadjoint matrices over the quaternions

Quaternion Matrix ◽

Complex Matrix ◽

The complex matrix representation for a quaternion matrix is used in this paper to find necessary and sufficient conditions for the existence of an $H$-selfadjoint $m$th root of a given $H$-selfadjoint quaternion matrix. In the process, when such an $H$-selfadjoint $m$th root exists, its construction is also given.

Least-Norm of the General Solution to Some System of Quaternion Matrix Equations and Its Determinantal Representations

Abstract and Applied Analysis ◽

10.1155/2019/9072690 ◽

2019 ◽

Vol 2019 ◽

pp. 1-18

Author(s):

Abdur Rehman ◽

Ivan Kyrchei ◽

Muhammad Akram ◽

Ilyas Ali ◽

Abdul Shakoor

Keyword(s):

General Solution ◽

Sufficient Conditions ◽

Matrix Equations ◽

Quaternion Matrix ◽

Skew Field ◽

Numerical Examples ◽

Cramer’S Rule ◽

We constitute some necessary and sufficient conditions for the system A1X1=C1, X1B1=C2, A2X2=C3, X2B2=C4, A3X1B3+A4X2B4=Cc, to have a solution over the quaternion skew field in this paper. A novel expression of general solution to this system is also established when it has a solution. The least norm of the solution to this system is also researched in this article. Some former consequences can be regarded as particular cases of this article. Finally, we give determinantal representations (analogs of Cramer’s rule) of the least norm solution to the system using row-column noncommutative determinants. An algorithm and numerical examples are given to elaborate our results.

A System of Periodic Discrete-time Coupled Sylvester Quaternion Matrix Equations

Algebra Colloquium ◽

10.1142/s1005386717000104 ◽

2017 ◽

Vol 24 (01) ◽

pp. 169-180 ◽

Cited By ~ 22

Author(s):

Zhuoheng He ◽

Qingwen Wang

Keyword(s):

General Solution ◽

Discrete Time ◽

Quaternion Algebra ◽

Sufficient Conditions ◽

Generalized Inverses ◽

Matrix Equations ◽

Quaternion Matrix ◽

Necessary And Sufficient ◽

Sylvester Matrix Equations

We in this paper derive necessary and sufficient conditions for the system of the periodic discrete-time coupled Sylvester matrix equations [Formula: see text] over the quaternion algebra to be consistent in terms of ranks and generalized inverses of the coefficient matrices. We also give an expression of the general solution to the system when it is solvable. The findings of this paper generalize some known results in the literature.

Hermitian Solutions to a Quaternion Matrix Equation

Applied Mechanics and Materials ◽

10.4028/www.scientific.net/amm.50-51.391 ◽

2011 ◽

Vol 50-51 ◽

pp. 391-395

Author(s):

Ning Li ◽

Jing Jiang ◽

Wen Feng Wang

Keyword(s):

Matrix Equation ◽

Quaternion Algebra ◽

Sufficient Conditions ◽

Quaternion Matrix ◽

Quaternion Matrix Equation ◽

The Common ◽

In this paper, we consider Hermitian and skew-Hermitian solutions to a certain matrix equation over quaternion algebra H. Necessary and sufficient conditions are obtained for the quaternion matrix equation to have Hermitian and skew-Hermitian solutions, and the expressions of such solutions are also given. As an application, the common skew-Hermitian g-inverse of quaternion matrix A and B is considered.

The Real and Complex Hermitian Solutions to a System of Quaternion Matrix Equations with Applications

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/2012/307036 ◽

2012 ◽

Vol 2012 ◽

pp. 1-19 ◽

Cited By ~ 1

Author(s):

Shao-Wen Yu

Keyword(s):

Classical System ◽

Sufficient Conditions ◽

Matrix Equations ◽

Quaternion Matrix ◽

The Real ◽

Necessary And Sufficient ◽

Real Matrices

We establish necessary and sufficient conditions for the existence of and the expressions for the general real and complex Hermitian solutions to the classical system of quaternion matrix equationsA1X=C1,XB1=C2, and A3XA3*=C3. Moreover, formulas of the maximal and minimal ranks of four real matricesX1,X2,X3, andX4in solutionX=X1+X2i+X3j+X4kto the system mentioned above are derived. As applications, we give necessary and sufficient conditions for the quaternion matrix equationsA1X=C1,XB1=C2,A3XA3*=C3, and A4XA4*=C4to have real and complex Hermitian solutions.

The extinction time of a general birth and death process with catastrophes

Journal of Applied Probability ◽

10.1017/s0021900200116031 ◽

1986 ◽

Vol 23 (04) ◽

pp. 851-858 ◽

Cited By ~ 1

Author(s):

P. J. Brockwell

Keyword(s):

Laplace Transform ◽

Size Distribution ◽

Sufficient Conditions ◽

Death Process ◽

Extinction Time ◽

Birth And Death Process ◽

Different Types ◽

The Laplace Transform ◽

The Laplace transform of the extinction time is determined for a general birth and death process with arbitrary catastrophe rate and catastrophe size distribution. It is assumed only that the birth rates satisfyλ0= 0,λj> 0 for eachj> 0, and. Necessary and sufficient conditions for certain extinction of the population are derived. The results are applied to the linear birth and death process (λj=jλ, µj=jμ) with catastrophes of several different types.

Does the Concatenated Generalized Matching Law Identify the Necessary and Sufficient Conditions?

PsycEXTRA Dataset ◽

10.1037/e537052012-028 ◽

2004 ◽

Author(s):

James S. MacDonall

Keyword(s):

Sufficient Conditions ◽

Matching Law ◽

Generalized Matching Law ◽

NECESSARY AND SUFFICIENT CONDITIONS OF EXISTENCE AND UNIQUENESS OF LIMIT CYCLES FOR A CLASS OF POLYNOMIAL SYSTEM

Acta Mathematica Scientia ◽

10.1016/s0252-9602(18)30430-2 ◽

1991 ◽

Vol 11 (1) ◽

pp. 65-71 ◽

Cited By ~ 1

Author(s):

Deming Liu

Keyword(s):

Limit Cycles ◽

Existence And Uniqueness ◽

Polynomial System ◽

Sufficient Conditions ◽

Uniqueness Of Limit Cycles ◽

Nonlinear boundary-value problems for degenerate differential-algebraic systems

Ukrainian Mathematical Bulletin ◽

10.37069/1810-3200-2020-17-3-2 ◽

2020 ◽

Vol 17 (3) ◽

pp. 313-324

Author(s):

Sergii Chuiko ◽

Ol'ga Nesmelova

Keyword(s):

Boundary Value Problem ◽

Boundary Value Problems ◽

Algebraic System ◽

Boundary Value ◽

Sufficient Conditions ◽

Weakly Nonlinear ◽

Differential Algebraic System ◽

Linear Differential ◽

The study of the differential-algebraic boundary value problems, traditional for the Kiev school of nonlinear oscillations, founded by academicians M.M. Krylov, M.M. Bogolyubov, Yu.A. Mitropolsky and A.M. Samoilenko. It was founded in the 19th century in the works of G. Kirchhoff and K. Weierstrass and developed in the 20th century by M.M. Luzin, F.R. Gantmacher, A.M. Tikhonov, A. Rutkas, Yu.D. Shlapac, S.L. Campbell, L.R. Petzold, Yu.E. Boyarintsev, V.F. Chistyakov, A.M. Samoilenko, O.A. Boichuk, V.P. Yacovets, C.W. Gear and others. In the works of S.L. Campbell, L.R. Petzold, Yu.E. Boyarintsev, V.F. Chistyakov, A.M. Samoilenko and V.P. Yakovets were obtained sufficient conditions for the reducibility of the linear differential-algebraic system to the central canonical form and the structure of the general solution of the degenerate linear system was obtained. Assuming that the conditions for the reducibility of the linear differential-algebraic system to the central canonical form were satisfied, O.A.~Boichuk obtained the necessary and sufficient conditions for the solvability of the linear Noetherian differential-algebraic boundary value problem and constructed a generalized Green operator of this problem. Based on this, later O.A. Boichuk and O.O. Pokutnyi obtained the necessary and sufficient conditions for the solvability of the weakly nonlinear differential algebraic boundary value problem, the linear part of which is a Noetherian differential algebraic boundary value problem. Thus, out of the scope of the research, the cases of dependence of the desired solution on an arbitrary continuous function were left, which are typical for the linear differential-algebraic system. Our article is devoted to the study of just such a case. The article uses the original necessary and sufficient conditions for the solvability of the linear Noetherian differential-algebraic boundary value problem and the construction of the generalized Green operator of this problem, constructed by S.M. Chuiko. Based on this, necessary and sufficient conditions for the solvability of the weakly nonlinear differential-algebraic boundary value problem were obtained. A typical feature of the obtained necessary and sufficient conditions for the solvability of the linear and weakly nonlinear differential-algebraic boundary-value problem is its dependence on the means of fixing of the arbitrary continuous function. An improved classification and a convergent iterative scheme for finding approximations to the solutions of weakly nonlinear differential algebraic boundary value problems was constructed in the article.