On some spectral properties of large self-dual dilute quaternion random matrices

In this paper, we study the spectral properties of the large self-dual dilute quaternion random matrices. For the dilute case, we prove that the empirical spectral distribution still converges to the semicircular law with some appropriate normalization. Further, we obtain the limits of the extreme eigenvalues of the large self-dual dilute quaternion random matrices under some moment assumptions of the underlying distributions and give a necessary condition for the strong convergence of the extreme eigenvalues.     

三维转动的四元数表述

用四元数表达三维的旋转与使用矩阵相比具有两个优点：第一．几何意义明确;第二,计算简单．因此,四元数在数学、物理学和计算机图形学中具有很高的应用价值．本文详细叙述了四元数的这一功能,讨论了四元数的定义、运算、性质、几何意义和它的3种表达形式,给出了使用四元数处理点的各种几何变换的一般结论．     

四元数矩阵方程AX=B的实部半正定解(英文)

一个ｎ×ｎ实四元数矩阵称为实部半正定（或正定）矩阵，如果对于任意的非零ｎ维四元数列向量ｘ，有Ｒｅ［ｘＡｘ］≥０（或＞０）．本文给出了四元数矩阵方程ＡＸ＝Ｂ有实部半正定（或正定）矩阵解的充要条件及其通解的表达式，并给出了四元数分块阵为实部半正定（或正定）矩阵的一个判别法则     

关于四元数自共轭矩阵乘积迹和特征值的几个定理

给出四元数自共轭矩阵乘积迹的几个定理及特征值之界的几个新估计，在四元数体上改进和推广了文［１－１２］的相应结果     

Controllability and Observability of Linear Quaternion-valued Systems

The aim of this paper is to define an extension of the controllability and observability for linear quaternion-valued systems (QVS). Some criteria for controllability and observability are derived, and the minimum norm control and duality theorem are also investigated. Compared with real-valued or complex-valued linear systems, it is shown that the classical Caylay-Hamilton Theorem as well as Popov-Belevitch-Hautus (PBH) type controllability and observability test do not hold for linear QVS. Hence, a modified PBH type necessary condition is studied for the controllability and observability, respectively. Finally, some examples are given to illustrate the effectiveness of the obtained results.     

Quadratic descent of hermitian forms

Quadratic descent of hermitian and skew hermitian forms over division algebras with involution of the first kind in arbitrary characteristic is investigated and a criterion, in terms of systems of quadratic forms, is obtained. A refined result is also obtained for hermitian (resp. skew hermitian) forms over a quaternion algebra with symplectic (resp. orthogonal) involution.     

Robust quaternion matrix completion with applications to image inpainting

In this paper, we study robust quaternion matrix completion and provide a rigorous analysis for provable estimation of quaternion matrix from a random subset of their corrupted entries. In order to generalize the results from real matrix completion to quaternion matrix completion, we derive some new formulas to handle noncommutativity of quaternions. We solve a convex optimization problem, which minimizes a nuclear norm of quaternion matrix that is a convex surrogate for the quaternion matrix rank, and the ?₁‐norm of sparse quaternion matrix entries. We show that, under incoherence conditions, a quaternion matrix can be recovered exactly with overwhelming probability, provided that its rank is sufficiently small and that the corrupted entries are sparsely located. The quaternion framework can be used to represent red, green, and blue channels of color images. The results of missing/noisy color image pixels as a robust quaternion matrix completion problem are given to show that the performance of the proposed approach is better than that of the testing methods, including image inpainting methods, the tensor‐based completion method, and the quaternion completion method using semidefinite programming.     

On metallic ratio in

Metallic ratio is a root of the simple quadratic equation x² = kx + 1 for k is any positive integer which is the characteristic equation of the recurrence relation of k‐Fibonacci (k‐Lucas) numbers. This paper is about the metallic ratio in . We define k‐Fibonacci and k‐Lucas numbers in , and we show that metallic ratio can be calculated in if and only if p≡ ± 1 mod (k² + 4), which is the generalization of the Gauss reciprocity theorem for any integer k. Also, we obtain that the golden ratio, the silver ratio, and the bronze ratio, the three together, can be calculated in for the first time. Moreover, we introduce k‐Fibonacci and k‐Lucas quaternions with some algebraic properties and some identities for them.     

Stability analysis of quaternion‐valued Cohen‐Grossberg neural networks

In this paper, by starting from basic quaternion algebra properties and algorithms, a quaternion‐valued Cohen‐Grossberg neural network was derived, subsequently, several new sufficient conditions are derived to ensure existence and global asymptotic stability (GAS) and global exponential stability (GES) of the equilibrium point (EP) for quaternion‐valued Cohen‐Grossberg neural networks. The obtained criteria can be checked easily in practice and have a distinguished feature from previous studies. Finally, we have numerical evidences that the mathematical system and the conclusions presented are validated.     

Quaternion <Emphasis Type="Italic">H</Emphasis>-type group and differential operator Δ<Subscript><Emphasis Type="Italic">λ</Emphasis></Subscript>

We study the relations between the quaternion H-type group and the boundary of the unit ball on the two-dimensional quaternionic space. The orthogonal projection of the space of square integrable functions defined on quaternion H-type group into its subspace of boundary values of q-holomorphic functions is considered. The precise form of Cauchy-Szegö kernel and the orthogonal projection operator is obtained. The fundamental solution for the operator Δ_λ is found.