Quaternion Matrix Research Articles

A Lyapunov-based three-axis attitude intelligent control approach for unmanned aerial vehicle

A novel Lyapunov-based three-axis attitude intelligent control approach via allocation scheme is considered in the proposed research to deal with kinematics and dynamics regarding the unmanned aerial vehicle systems. There is a consensus among experts of this field that the new outcomes in the present complicated systems modeling and control are highly appreciated with respect to state-of-the-art. The control scheme presented here is organized in line with a new integration of the linear-nonlinear control approaches, as long as the angular velocities in the three axes of the system are accurately dealt with in the inner closed loop control. And the corresponding rotation angles are dealt with in the outer closed loop control. It should be noted that the linear control in the present outer loop is first designed through proportional based linear quadratic regulator (PD based LQR) approach under optimum coefficients, while the nonlinear control in the corresponding inner loop is then realized through Lyapunov-based approach in the presence of uncertainties and disturbances. In order to complete the inner closed loop control, there is a pulse-width pulse-frequency (PWPF) modulator to be able to handle on-off thrusters. Furthermore, the number of these on-off thrusters may be increased with respect to the investigated control efforts to provide the overall accurate performance of the system, where the control allocation scheme is realized in the proposed strategy. It may be shown that the dynamics and kinematics of the unmanned aerial vehicle systems have to be investigated through the quaternion matrix and its corresponding vector to avoid presenting singularity of the results. At the end, the investigated outcomes are presented in comparison with a number of potential benchmarks to verify the approach performance.

The Spectral Theorem for Unitary Operators Based on the S-Spectrum

The quaternionic spectral theorem has already been considered in the literature, see e.g. [22], [31], [32], however, except for the finite dimensional case in which the notion of spectrum is associated to an eigenvalue problem, see [21], it is not specified which notion of spectrum underlies the theorem. In this paper we prove the quaternionic spectral theorem for unitary operators using the $S$-spectrum. In the case of quaternionic matrices, the $S$-spectrum coincides with the right-spectrum and so our result recovers the well known theorem for matrices. The notion of $S$-spectrum is relatively new, see [17], and has been used for quaternionic linear operators, as well as for $n$-tuples of not necessarily commuting operators, to define and study a noncommutative versions of the Riesz-Dunford functional calculus. The main tools to prove the spectral theorem for unitary operators are the quaternionic version of Herglotz's theorem, which relies on the new notion of $q$-positive measure, and quaternionic spectral measures, which are related to the quaternionic Riesz projectors defined by means of the $S$-resolvent operator and the $S$-spectrum. The results in this paper restore the analogy with the complex case in which the classical notion of spectrum appears in the Riesz-Dunford functional calculus as well as in the spectral theorem.

Iterative algorithms for least-squares solutions of a quaternion matrix equation

This paper deals with developing four efficient algorithms (including the conjugate gradient least-squares, least-squares with QR factorization, least-squares minimal residual and Paige algorithms) to numerically find the (least-squares) solutions of the following (in-) consistent quaternion matrix equation $$\begin{aligned} {A_1}X + {\left( {{A_1}X} \right) ^{\eta H}} + {B_1}YB_1^{\eta H} + {C_1}ZC_1^{\eta H} = {D_1}, \end{aligned}$$ in which the coefficient matrices are large and sparse. More precisely, we construct four efficient iterative algorithms for determining triple least-squares solutions (X, Y, Z) such that X may have a special assumed structure, Y and Z can be either $$\eta $$ -Hermitian or $$\eta $$ -anti-Hermitian matrices. In order to speed up the convergence of the offered algorithms for the case that the coefficient matrices are possibly ill-conditioned, a preconditioned technique is employed. Some numerical test problems are examined to illustrate the effectiveness and feasibility of presented algorithms.

A system of generalized Sylvester quaternion matrix equations and its applications

Let Hm×n be the set of all m × n matrices over the real quaternion algebra. We call that A∈Hn×n is η-Hermitian if A=−ηA*η,η ∈ {i, j, k}, where i, j, k are the quaternion units. Denote Aη*=−ηA*η. In this paper, we derive some necessary and sufficient conditions for the solvability to the system of generalized Sylvester real quaternion matrix equations AiXi+YiBi+CiZDi=Ei,(i=1,2), and give an expression of the general solution to the above mentioned system. As applications, we give some solvability conditions and general solution for the generalized Sylvester real quaternion matrix equation A1X+YB1+C1ZD1=E1, where Z is required to be η-Hermitian. We also present some solvability conditions and general solution for the system of real quaternion matrix equations involving η-Hermicity AiXi+(AiXi)η*+BiYBiη*=Ci,(i=1,2), where Y is required to be η-Hermitian. Our results include some well-known results as special cases.

A system of matrix equations with five variables

In this paper, we give some necessary and sufficient conditions for the consistence of the system of quaternion matrix equationsA1X=C1,YB1=D1,A2W=C2,ZB2=D2,A3V=C3,VB3=C4,A4VB4=C5,A5X+YB5+C6W+ZD6+E6VF6=G6,and constitute an expression of the general solution to the system when it is solvable. The outcomes of this paper encompass some recognized results in the collected works. In addition, we establish an algorithm and a numerical example to illustrate the theory constructed in the paper.

Fundamentals of Quaternionic Kinematics in Euclidean 4-Space

Based on a kinematic mapping for the group SE(4) of displacements of Euclidean 4-space, we show that the mapping of basic elements (points, oriented lines, oriented planes, oriented hyperplanes, instantaneous screws) can be written compactly in terms of 2 × 2 quaternionic matrices. Moreover we discuss the kinematics on the velocity level by investigating instantaneous screws and their geometric parameters.

Consimilarity of quaternions and coneigenvalues of quaternion matrices

First of all, by characterizing solutions of the quaternion equation ax=x˜b, this paper studies consimilarity of quaternions and some related consequences. For the important role of coneigenvalues in consimilarity tranformations of quaternion matrices, this paper further derives the relations between principle right coneigenvalues of a quaternion matrix and eigenvalues of the corresponding real representation matrix. Then, based on the real representation matrix, an effective algorithm is presented to calculate all coneigenvalues and the associated coneigenvectors of a quaternion matrix. Finally, two numerical examples are given to verify the effectiveness of the proposed algorithm.

Matric variate Pearson type II-Riesz distribution

The Pearson type II distribution is well known and is used in the general framework of real normed division algebras and Riesz distribution theory. Also, the so called Pearson type II-Riesz distribution, based on the Kotz–Riesz distribution, is presented in a unified way valid in the context of real, complex, quaternion and octonion random matrices. Specifically, the central nonsingular matric variate generalised Pearson type II-Riesz distribution and beta-Riesz type I distributions are derived in the addressed multiple numerical field settings.

Algebraic methods for least squares problem in split quaternionic mechanics

A split quaternionic least squares problem is one method of solving overdetermined sets of split quaternion linear equations AX ≈ B that is appropriate when there is error in the matrix B. In this paper, by means of complex representation and real representation of a split quaternion matrix, we study the split quaternionic least squares problem, derive two algebraic methods for finding solutions of the problems in split quaternionic mechanics.

Special least squares solutions of the quaternion matrix equation [formula omitted] with applications

In this paper, by applying particular structure of the real representations of quaternion matrices and the Moore–Penrose generalized inverse, we derive the expressions of the minimal norm least squares solution, the pure imaginary least squares solution, and the real least squares solution for the quaternion matrix equation AX=B. The resulting formulas only involve real matrices, which are simpler than those reported in (Yuan et al., 2013). The corresponding algorithms only perform real arithmetic which also consider particular structure of the real representations of quaternion matrices, therefore are very efficient and easily understood. Numerical examples are provided to illustrate the efficiency of our algorithms.

An algebraic technique for total least squares problem in quaternionic quantum theory

The total least squares (TLS) is a method of fitting that is appropriate when there are errors in both the observation vector b = b m × 1 and the data matrix A = A m × n . In this paper, we study the quaternion total least squares (QTLS) problem by means of real representations of quaternion matrices, and derive an algebraic technique for finding solutions of the QTLS problem in quaternionic quantum theory.

Algebraic techniques for diagonalization of a split quaternion matrix in split quaternionic mechanics

In the study of the relation between complexified classical and non-Hermitian quantum mechanics, physicists found that there are links to quaternionic and split quaternionic mechanics, and this leads to the possibility of employing algebraic techniques of split quaternions to tackle some problems in complexified classical and quantum mechanics. This paper, by means of real representation of a split quaternion matrix, studies the problem of diagonalization of a split quaternion matrix and gives algebraic techniques for diagonalization of split quaternion matrices in split quaternionic mechanics.

Two Novel Algebraic Techniques for Quaternion Least Squares Problems in Quaternionic Quantum Mechanics

Quaternion least squares (QLS) problem is one method of solving overdetermined sets of quaternion linear equations \({AX \approx B}\) and \({AXC \approx B}\) that are appropriate when there are errors in the matrix B. This paper, by means of complex representation and real representation of a quaternion matrix, studies the QLS problems, and derives two new algebraic techniques for finding solutions of the QLS problem in quaternionic quantum mechanics.

Solution to a system of real quaternion matrix equations encompassing η-Hermicity

Let Hm×n be the set of all m × n matrices over the real quaternion algebra H={c0+c1i+c2j+c3k∣i2=j2=k2=ijk=−1,c0,c1,c2,c3∈R}. A∈Hn×n is known to be η-Hermitian if A=Aη*=−ηA*η,η∈{i,j,k} and A* means the conjugate transpose of A. We mention some necessary and sufficient conditions for the existence of the solution to the system of real quaternion matrix equations including η-HermicityA1X=C1,A2Y=C2,YB2=D2,Y=Yη*,A3Z=C3,ZB3=D3,Z=Zη*,A4X+(A4X)η*+B4YB4η*+C4ZC4η*=D4,and also construct the general solution to the system when it is consistent. The outcome of this paper diversifies some particular results in the literature. Furthermore, we constitute an algorithm and a numerical example to comprehend the approach established in this treatise.

Spectrahedral cones generated by rank 1 matrices

Let $$\mathcal{S}_+^n \subset \mathcal{S}^n$$S+n?Sn be the cone of positive semi-definite matrices as a subset of the vector space of real symmetric $$n \times n$$n?n matrices. The intersection of $$\mathcal{S}_+^n$$S+n with a linear subspace of $$\mathcal{S}^n$$Sn is called a spectrahedral cone. We consider spectrahedral cones K such that every element of K can be represented as a sum of rank 1 matrices in K. We shall call such spectrahedral cones rank one generated (ROG). We show that ROG cones which are linearly isomorphic as convex cones are also isomorphic as linear sections of the positive semi-definite matrix cone, which is not the case for general spectrahedral cones. We give many examples of ROG cones and show how to construct new ROG cones from given ones by different procedures. We provide classifications of some subclasses of ROG cones, in particular, we classify all ROG cones for matrix sizes not exceeding 4. Further we prove some results on the structure of ROG cones. We also briefly consider the case of complex or quaternionic matrices. ROG cones are in close relation with the exactness of semi-definite relaxations of quadratically constrained quadratic optimization problems or of relaxations approximating the cone of nonnegative functions in squared functional systems.

L-structured quaternion matrices and quaternion linear matrix equations

This paper focuses on L-structured quaternion matrices. L-structured real matrices, conditions for the existence of solutions and the general solution of linear matrix equations were studied in the paper [Magnus JR. L-structured matrices and linear matrix equations, Linear Multilinear Algebra 1983;14:67–88]. In this paper, we present a theoretical study extending L-structured real matrices to L-structured quaternion matrices, and introduce some L-structured quaternion matrices. Based on them, we then discuss their applications in quaternion matrix equations.

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.

Split Quaternion Matrix Representation of Dual Split Quaternions and Their Matrices

In this paper, we present some important properties of matrices over dual split quaternions. We examine dual split quaternion matrices by their split quaternion matrix representation. Then, we give some interesting results for 2 × 2 split quaternion matrix representation of a dual split quaternion. Finally, we prove that the group \({\mathcal{H}_{\mathbb{D}}}\) and \({SC(2, \widehat{\mathbb{H}})}\) are isomorphic.

ELEMENTARY FUNCTIONS OF A QUATERNION VARIABLE AND SOME APPLICATIONS

In this paper, an effective formula for the calculation of the elementary functions of a quaternion variable obtained using the methods of differential equations. Also the elementary functions are obtained from the quaternion matrices.

On the Semicircular Law of Large-Dimensional Random Quaternion Matrices

It is well known that the Gaussian symplectic ensemble is defined on the space of \(n\times n\) quaternion self-dual Hermitian matrices with Gaussian random elements. There is a huge body of literature regarding this kind of matrices based on the exact known form of the density function of the eigenvalues (see Erdős in Russ Math Surv 66(3):507–626, 2011; Erdős in Probab Theory Relat Fields 154(1–2):341–407, 2012; Erdős et al. in Adv Math 229(3):1435–1515, 2012; Knowles and Yin in Probab Theory Relat Fields, 155(3–4):543–582, 2013; Tao and Vu in Acta Math 206(1):127–204, 2011; Tao and Vu in Electron J Probab 16(77):2104–2121, 2011). Due to the fact that multiplication of quaternions is not commutative, few works about large-dimensional quaternion self-dual Hermitian matrices are seen without normality assumptions. As in natural, we shall get more universal results by removing the Gaussian condition. For the first step, in this paper, we prove that the empirical spectral distribution of the common quaternion self-dual Hermitian matrices tends to the semicircular law. The main tool to establish the universal result is given as a lemma in this paper as well.