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A Real Representation Method for Solving Yakubovich- -Conjugate Quaternion Matrix Equation  [PDF]
Caiqin Song,Jun-e Feng,Xiaodong Wang,Jianli Zhao
Abstract and Applied Analysis , 2014, DOI: 10.1155/2014/285086
Abstract: A new approach is presented for obtaining the solutions to Yakubovich- -conjugate quaternion matrix equation based on the real representation of a quaternion matrix. Compared to the existing results, there are no requirements on the coefficient matrix . The closed form solution is established and the equivalent form of solution is given for this Yakubovich- -conjugate quaternion matrix equation. Moreover, the existence of solution to complex conjugate matrix equation is also characterized and the solution is derived in an explicit form by means of real representation of a complex matrix. Actually, Yakubovich-conjugate matrix equation over complex field is a special case of Yakubovich- -conjugate quaternion matrix equation . Numerical example shows the effectiveness of the proposed results. 1. Introduction The linear matrix equation , which is called the Kalman-Yakubovich matrix equation in [1], is closely related to many problems in conventional linear control systems theory, such as pole assignment design [2], Luenberger-type observer design [3, 4], and robust fault detection [5, 6]. In recent years, many studies have been reported on the solutions to many algebraic equations including quaternion matrix equations and nonlinear matrix equations. Yuan and Liao [7] investigated the least squares solution of the quaternion -conjugate matrix equation (where denotes the -conjugate of quaternion matrix ) with the least norm using the complex representation of quaternion matrix, the Kronecker product of matrices, and the Moore-Penrose generalized inverse. The authors in [8] considered the matrix nearness problem associated with the quaternion matrix equation by means of the CCD-Q, GSVD-Q, and the projection theorem in the finite dimensional inner product space. In addition, Song et al. [9, 10] established the explicit solutions to the quaternion -conjugate matrix equation , , but here the known quaternion matrix is a block diagonal form. Wang et al. in [11, 12] investigated Hermitian tridiagonal solutions and the minimal-norm solution with the least norm of quaternionic least squares problem in quaternionic quantum theory. Besides, in [13, 14], some solutions for the Kalman-Yakubovich equation are presented in terms of the coefficients of characteristic polynomial of matrix or the Leverrier algorithm. The existence of solution to the matrix equation , which, for convenience, is called the Kalman-Yakubovich-conjugate matrix equation, is established, and the explicit solution is derived. Several necessary and sufficient conditions for the existence of a unique
Functions of several quaternion variables and quaternion manifolds  [PDF]
S. V. Ludkovsky
Mathematics , 2003,
Abstract: Functions of several quaternion variables are investigated and integral representation theorems for them are proved. With the help of them solutions of the $\tilde \partial $-equations are studied. Moreover, quaternion Stein manifolds are defined and investigated.
Monotone trace functions of several variables  [PDF]
Frank Hansen
Mathematics , 2006,
Abstract: We investigate monotone operator functions of several variables under a trace or a trace-like functional. In particular, we prove the inequality \tau(x_1... x_n)\le\tau(y_1... y_n) for a trace \tau on a C^*-algebra and abelian n-tuples (x_1,...,x_n)\le (y_1,...,y_n) of positive elements. We formulate and prove Jensen's inequality for expectation values, and we study matrix functions of several variables which are convex or monotone with respect to the weak majorization for matrices.
Trace theorems: critical cases and best constants  [PDF]
Michael Ruzhansky,Mitsuru Sugimoto
Mathematics , 2012, DOI: 10.1090/S0002-9939-2014-12207-6
Abstract: The purpose of this paper is to present the critical cases of the trace theorems for the restriction of functions to closed surfaces, and to give the asymptotics for the norms of the traces under dilations of the surface. We also discuss the best constants for them.
On symmetry of extremals in several embedding theorems  [PDF]
E. V. Mukoseeva,A. I. Nazarov
Mathematics , 2014,
Abstract: We study the symmetry/asymmetry of functions providing sharp constants in the embedding theorems ${\stackrel{\circ}{W}}\vphantom{W}_2^r(-1,1)\hookrightarrow{\stackrel{\circ}{W}}\vphantom{W}_\infty^k(-1,1)$ for various $r$ and $k$. The sharp constants for all $r>k$ in the cases $k=4$ and $k=6$ are calculated explicitly as well.
Comparison theorems for conjugate points in sub-Riemannian geometry  [PDF]
Davide Barilari,Luca Rizzi
Mathematics , 2014, DOI: 10.1051/cocv/2015013
Abstract: We prove sectional and Ricci-type comparison theorems for the existence of conjugate points along sub-Riemannian geodesics. In order to do that, we regard sub-Riemannian structures as a special kind of variational problems. In this setting, we identify a class of models, namely linear quadratic optimal control systems, that play the role of the constant curvature spaces. As an application, we prove a version of sub-Riemannian Bonnet-Myers theorem and we obtain some new results on conjugate points for three dimensional left-invariant sub-Riemannian structures.
Jensen's trace inequality in several variables  [PDF]
Frank Hansen,Gert K. Pedersen
Mathematics , 2003,
Abstract: Jensen's trace inequality is established for every multivariable, convex function and every trace or trace-like functional on a C*-algebra.
Some Operations on Quaternion Numbers
Bo Li, Xiquan Liang, Pan Wang, Yanping Zhuang
Formalized Mathematics , 2009, DOI: 10.2478/v10037-009-0006-x
Abstract: In this article, we give some equality and basic theorems about quaternion numbers, and some special operations.
Bernstein Type Theorems For Minimal Lagrangian Graphs of Quaternion Euclidean space  [PDF]
Yuxin Dong,Yingbo Han,Qingchun Ji
Mathematics , 2006,
Abstract: In this paper, we prove some Bernstein type results for $n$-dimensional minimal Lagrangian graphs in quaternion Euclidean space $H^n\cong R^{4n}$. In particular, we also get a new Bernstein Theorem for special Lagrangian graphs in $C^n$
Quaternion Involutions  [PDF]
Todd A. Ell,Stephen J. Sangwine
Mathematics , 2005, DOI: 10.1016/j.camwa.2006.10.029
Abstract: An involution is usually defined as a mapping that is its own inverse. In this paper, we study quaternion involutions that have the additional properties of distribution over addition and multiplication. We review formal axioms for such involutions, and we show that the quaternions have an infinite number of involutions. We show that the conjugate of a quaternion may be expressed using three mutually perpendicular involutions. We also show that any set of three mutually perpendicular quaternion involutions is closed under composition. Finally, we show that projection of a vector or quaternion can be expressed concisely using involutions.
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