四元数非中心x^2分布，t分布，F分布及性质

补充四元数线性变换下四元数正态分布的性质,给出四元数非中心X^2分布、t分布,F分布的定义,导出密度函数及其性质,并研究四元数正态分布条件下样本均值及方差的分布。     

四元数Beta分布、F分布及其特征根分布

本文利用外积、外微分式运算作工具，推导出四元数Beta分布、F分布及其特征根分布的密度函数。     

四元数非中心χ~2分布,t分布,F分布及性质

补充四元数线性变换下四元数正态分布的性质 ,给出四元数非中心 χ2 分布 ,t分布 ,F分布的定义 ,导出密度函数及其性质 ,并研究四元数正态分布条件下样本均值及方差的分布 .     

四元数非中心Wishart分布及其特征分布

利用四元数矩阵变元的带状多项式定义及性质推导出四元数非中心Wishart分布及其特征根分布。     

一类四元数小波包的构造

实值二维信号可以用四元数来表示,因此,四元数的尺度函数和小波的构造就成为分析二维信号的关键．引入了四元数小波包的概念,并且借助于四元数多分辨分析和四元数尺度函数和四元数小波函数的概念和若干公式,给出并构造了一类四元数正交小波包的构造方法,得到了四元数正交小波包的3个正交性公式,最后,利用四元数正交小波包给出了L^2（R...     

四元数向量和矩阵的秩

在四元数和四元数向量、矩阵空间上引入三种不同的实表示法则，将四元数列向量的左右线性相关性问题转换成实数域上向量的线性相关问题，由此获得用实矩阵的秩代替四元数矩阵列左秩和列右秩计算方法，同时得出四元数矩阵可逆的一些充要条件和一些新的四元数行列式定义．     

奇异四元数矩阵的正态及Wishart分布

本文利用拉直算子(vec)和Kronecker积求得了四元数矩阵的实表示矩阵的一些性质,在此基础上利用实矩阵的奇异正态分布密度函数,求出了四元数矩阵的奇异正态分布的密度函数表达式.由此得到四元数矩阵奇异Wishart分布的密度函数表达式.     

四元数体上矩阵的广义对角化

引入了复四元数环和四元数体上矩阵可 对角化的概念,研究了复四元数环上矩阵的性质,给出了四元数体上矩阵可 对角化的充分必要条件和求矩阵 对角化的方法。     

分裂四元数矩阵的奇异值分解及应用

利用i-共轭重新定义了分裂四元数矩阵的共轭转置,在此基础上借助复表示和友向量研究了分裂四元数矩阵的奇异值分解,并利用所得结果解决了分裂四元数矩阵的极分解和分裂四元数矩阵方程AXB-CYD=E.     

关于四元数矩阵之迹的几个定理

R.Rellman对两个正定实矩阵建立了与Cauchy—Schwarz不等式相类似的结果,引起人们的关注,对Rellman不等式进行深入的研究.但对四元数矩阵之迹的研究至今未见.如所熟知,四元数体的非交换性,已经给四元数代数理论的研究带来了巨大的困难,它也必然影响到四元数矩阵迹的性质.事实上,关于实(或复)矩阵的几个简单性质:     

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.     

The maximal and minimal ranks of a quaternion matrix expression with applications

In this paper, we establish the formulas of the extermal ranks of the quaternion matrix expression f(X₁, X₂) = C₇ ? A₄X₁B₄ ? A₅X₂B₅ where X₁, X₂ are variant quaternion matrices subject to quaternion matrix equations A₁X₁ = C₁, A₂X₁ = C₂, A₃X₁ = C₃, X₂B₁ = C₄, X₂B₂ = C₅, X₂B₃ = C₆. As applications, we give a new necessary and sufficient condition for the existence of solutions to some systems of quaternion matrix equations. Some results can be viewed as special cases of the results of this paper.     

Quaternion algebras with derivations

We study derivations on quaternion algebras that stabilise quadratic subfields. Following the work of L. Juan and A. Magid [10], we provide an explicit construction of a differential splitting field for a given differential quaternion algebra. We also examine the presence and impact of those derivations on quaternion algebras that admit new constants.     

Split Quaternions and Integer-valued Polynomials

The integer split quaternions form a noncommutative algebra over ?. We describe the prime and maximal spectrum of the integer split quaternions and investigate integer-valued polynomials over this ring. We prove that the set of such polynomials forms a ring, and proceed to study its prime and maximal ideals. In particular we completely classify the primes above 0, we obtain partial characterizations of primes above odd prime integers, and we give sufficient conditions for building maximal ideals above 2.     

The numerical radius and specttural matrices

In this paper we investigate spectral matrices, i.e., matrices with equal spectral and numerical radii. Various characterizations and properties of these matrices are given.     

Quaternionic Numerical Range and Real Subspaces

Let W(A) be the numerical range of an n × n quaternionic matrix A and V a real subspace of the skew field of real quaternions. In this note the authors consider the relation among the shape of W(A), the convexity of V∩W(A): and the validity of the equality V∩W(A) = W_v(A), where W_v (A) is the orthogonal projection of W(A) into V.     

四元数矩阵凸函数及其不等式

本文推广复矩阵凸函数及其性质到四元数体上矩阵，并且得到关于四元数矩阵凸函数的一些不等式．     

Two-dimensional quaternion wavelet transform

In this paper we introduce the continuous quaternion wavelet transform (CQWT). We express the admissibility condition in terms of the (right-sided) quaternion Fourier transform. We show that its fundamental properties, such as inner product, norm relation, and inversion formula, can be established whenever the quaternion wavelets satisfy a particular admissibility condition. We present several examples of the CQWT. As an application we derive a Heisenberg type uncertainty principle for these extended wavelets.