Algebraic methods for equality constrained least squares problems in commutative quaternionic theory

This paper, by means of two matrix representations of a commutative quaternion matrix, studies the relationship between the solutions of commutative quaternion equality constrained least squares (LSE) problems and that of complex and real LSE problems and derives two algebraic methods for finding the solutions of equality constrained least squares problems in commutative quaternionic theory.     

Algebraic Method for Inequality Constrained Quaternion Least Squares Problem

In this paper, we introduce a kind of complex representation of quaternion matrices (or quaternion vectors) and quaternion matrix norms, study quaternionic least squares problem with quadratic inequality constraints (LSQI) by means of generalized singular value decomposition of quaternion matrices (GSVD), and derive a practical algorithm for finding solutions of the quaternionic LSQI problem in quaternionic quantum theory.     

广义四元数的一次代数方程

本文运用广义四元数代数的矩阵表示讨论了两类广义四元数的一次代数方程的解问题，并得到了这两类代数方程有唯一解、无穷多解，无解的判别条件。     

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.     

Real matrix representations of complex split quaternions with applications

In this paper, we introduce 8×8 real matrix representations of complex split quaternions. Then, the relations between real matrix representations of split and complex split quaternions are stated. Moreover, we investigate some linear split and complex split quaternionic equations with split Fibonacci and complex split Fibonacci quaternion coefficients. Finally, we also give some numerical examples as applications of real matrix representation of complex split quaternions.     

Sharper uncertainty principles in quaternionic Hilbert spaces

The uncertainty principle for quaternionic linear operators in quaternionic Hilbert spaces is established, which generalizes the result of Goh-Micchelli. It turns out that there appears an additional term given by a commutator that reflects the feature of quaternions. The result is further strengthened when one operator is self-adjoint, which extends under weaker conditions the uncertainty principle of Dang-Deng-Qian from complex numbers to quaternions. In particular, our results are applied to concrete settings related to quaternionic Fock spaces, quaternionic periodic functions, quaternion Fourier transforms, quaternion linear canonical transforms, and nonharmonic quaternion Fourier transforms.     

四元数矩阵方程(AXB,CXD)=(E,F)的最小范数最小二乘Hermitian解

提出了研究四元数矩阵方程(AXB, CXD)=(E, F)的最小范数最小二乘Hermitian解的一个有效方法.首先应用四元数矩阵的实表示矩阵以及实表示矩阵的特殊结构,把四元数矩阵方程转化为相应的实矩阵方程,然后求出四元数矩阵方程(AXB, CXD)=(E, F)的最小二乘Hermitian解集,进而得到其最小范数最小二乘Hermitian解.所得到的结果只涉及实矩阵,相应的算法只涉及实运算,因此非常有效.最后的两个数值例子也说明了这一点.     

Two-sided linear split quaternionic equations with n unknowns

In this paper, we investigate linear split quaternionic equations with the terms of the form axb. We give a new method of solving general linear split quaternionic equations with one, two and n unknowns. Moreover, we present some examples to show how this procedure works.     

Uncertainty principles associated with quaternionic linear canonical transforms

In the present paper, we generalize the linear canonical transform (LCT) to quaternion‐valued signals, known as the quaternionic LCT (QLCT). Using the properties of the LCT, we establish an uncertainty principle for the two‐sided QLCT. This uncertainty principle prescribes a lower bound on the product of the effective widths of quaternion‐valued signals in the spatial and frequency domains. It is shown that only a Gaussian quaternionic signal minimizes the uncertainty. Copyright © 2016 John Wiley & Sons, Ltd.     

A new technique of quaternion equality constrained least squares problem

By means of complex representation of a quaternion matrix, we study the relationship between the solutions of the quaternion equality constrained least squares problem and that of complex equality constrained least squares problem, and obtain a new technique of finding a solution of the quaternion equality constrained least squares problem.     

四元数广义奇异值分解及其应用

This paper derives a theorem of generalized singular value decomposition of quaternion matrices(QGSVD),studies the solution of general quaternion matrix equation AXB-CYD=E,and obtains quaternionic Roth's theorem.This paper also suggestssufficient and necessary conditions for the existence and uniqueness of solutions and explicit forms of the solutions of the equation.     

QUATERNION GENERALIZED SINGULAR VALUE DECOMPOSITION AND ITS APPLICATIONS

This paper derives a theorem of generalized singular value decomposition of quaternion matrices （QGSVD）,studies the solution of general quaternion matrix equation AXB -CYD= E,and obtains quaternionic Roth＇s theorem. This paper also suggests sufficient and necessary conditions for the existence and uniqueness of solutions and explicit forms of the solutions of the equation.     

关于四元数代数同构的条件

引入广义四元数代数的 K上表示矩阵的概念 ,探讨复线性表示与 K上表示矩阵的关系 .在此基础上 ,给出两个广义四元数代数同构的一个新的判定条件     

关于“四元数分析中的Neumann边值问题”的注记

本文指出了文［１］中的几个错误,并利用?－方程解的积分表示,获得了其所讨论的超球上的四元数方程的Ｎｅｕｍａｎｎ问题解的积分表示     

Equality constrained least squares problem over quaternion field

By using complex representation and GSVD of quaternion matrices, we define the norm of quaternion matrices, study the equality constrained least squares problem of quaternion matrices, give the necessary and sufficient conditions for the quaternion equality constrained least squares problem to have solutions, and finally derive a practical algorithm.     

Weierstrass method for quaternionic polynomial root‐finding

Quaternions, introduced by Hamilton in 1843 as a generalization of complex numbers, have found, in more recent years, a wealth of applications in a number of different areas that motivated the design of efficient methods for numerically approximating the zeros of quaternionic polynomials. In fact, one can find in the literature recent contributions to this subject based on the use of complex techniques, but numerical methods relying on quaternion arithmetic remain scarce. In this paper, we propose a Weierstrass‐like method for finding simultaneously all the zeros of unilateral quaternionic polynomials. The convergence analysis and several numerical examples illustrating the performance of the method are also presented.     

Generalised connected sums of quaternionic manifolds

Using deformations of singular twistor spaces, a generalisation of the connected sum construction appropriate for quaternionic manifolds is introduced. This is used to construct examples of quaternionic manifolds which have no quaternionic symmetries and leads to examples of quaternionic manifolds whose twistor spaces have arbitrary algebraic dimension.Partially supported by the National Science Foundation grant DMS-9296168.     

Real hypersurfaces in quaternionic projective space

Summary This paper is devoted to make a systematic study of real hypersurfaces of quaternionic projective space using focal set theory. We obtain three types of such real hypersurfaces. Two of them are known. Third type is new and in its study the first example of proper quaternion CR-submanifold appears. We study real hypersurfaces with constant principal curvatures and classify such hypersurfaces with at most two distinct principal curvatures. Finally we study the Ricci tensor of a real hypersurface of quaternionic projective space and classify pseudo-Einstein, almost-Einstein and Einstein real hypersurfaces.     

Algebraic Methods for Condiagonalization Under Consimilarity of Quaternion Matrices in Quaternionic Quantum Mechanics

By means of complex representation and real representation of quaternion matrices, paper [7] studied the problem of diagonalization of quaternion matrices. This paper introduces two new complex representation and real representation of quaternion matrices, studies the problem of condiagonalization under consimilarity of quaternion matrices, and gives two algebraic methods for the condiagonalization under consimilarity of quaternion matrices.     

A quaternionic proof of the representation formula of a quaternary quadratic form

The celebrated Four Squares Theorem of Lagrange states that every positive integer is the sum of four squares of integers. Interest in this Theorem has motivated a number of different demonstrations. While some of these demonstrations prove the existence of representations of an integer as a sum of four squares, others also produce the number of such representations. In one of these demonstrations, Hurwitz was able to use a quaternion order to obtain the formula for the number of representations. Recently the author has been able to use certain quaternion orders to demonstrate the universality of other quaternary quadratic forms besides the sum of four squares. In this paper, we develop results analogous to Hurwitz's above mentioned work by delving into the number theory of one of these quaternion orders, and discover an alternate proof of the representation formula for the corresponding quadratic form.