Yongge Tian
We present in this paper some fundamental tools for developing matrix analysis over the complex quaternion algebra. As applications, we consider generalized inverses, eigenvalues and eigenvectors, similarity, determinants of complex quaternion matrices, and so on.
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In this paper, we introduce the generalized Fibonacci-Lucas quaternions and we prove that the set of these elements is an order,in the sense of ring theory, of a quaternion algebra. Moreover, we investigate some properties of these elements.
In this paper we find a Clifford algebra associated to generalized Fibonacci quaternions. In this way, we provide a nice algorithm to obtain a division quaternion algebra starting from a quaternion non-division algebra and vice-versa.
The theory of the column-row determinants has been considered for matrices over a non-split quaternion algebra. In this paper the concepts of column-row determinants are extending to a split quaternion algebra. New definitions of the column and row immanants (permanents) for matrices over a non-split quaternion algebra are introduced, and their basic properties are investigated. The key theorem about the column and row immanants of a Hermitian matrix over a split quaternion algebra is proved. Based on this theorem an immanant of a Hermitian matrix over a split quaternion algebra is introduced.
The paper presents a classification of quadratic extension algebras, also known as algebras of degree 2, as well as several characterizations of quaternion algebras over a field (of characteristic not 2). The presentation is not restricted to finite-dimensional algebras. The 'pure calculus' on a quaternion algebra is introduced; it generalizes the 'vector calculus' of the Hamilton's quaternions, and is instrumental in establishing, in a clear and simple way, the isomorphism between the group consisting of all automorphisms and all anti-automorphisms of a quaternion algebra and the orthogonal g…
Quaternions often appear in wide areas of applied science and engineering such as wireless communications systems, mechanics, etc. It is known that are two types of non-isomorphic generalized quaternion algebras, namely: the algebra of quaternions and the algebra of coquaternions. In this paper, we present the formulae to pass from a basis in the generalized quaternion algebras to a basis in the division quaternions algebra or to a basis in the coquaternions algebra and vice versa. The same result was obtained for the generalized octonion algebra. Moreover, we emphasize the applications of the…
Starting from known results, due to Y. Tian in [Ti; 00], referring to the real matrix representations of the real quaternions, in this paper we will investigate the left and right real matrix representations for the complex quaternions and we give some examples in the special case of the complex Fibonacci quaternions.
The question of whether a split tensor product of quaternion algebras with involution over a field of characteristic two can be expressed as a tensor product of split quaternion algebras with involution, is shown to have an affirmative answer.
Splitting of differential quaternion algebras
本文研究带导子的四元数代数的微分分裂域问题，指出其未必在代数扩张上分裂，并利用 Riccati 方程的解给出分裂域超越次数的界。
Cramer's rules for some left, right and two-sided quaternion matrix equations are obtained within the framework of the theory of the column and row determinants.
The Rota-Baxter algebra structures on split semi-quaternion algebra
本文研究分裂半四元数代数上的Rota-Baxter算子，给出其矩阵刻画与所有权重下所有算子的矩阵表示，并证明Ma等人关于Sweedler代数的相关结果是本文结论的特例。
Algebras of Toeplitz Matrices with Quaternion Entries
本文针对四元数 Toeplitz 矩阵的极大左代数分类难题，引入了若干以四元数代数为元素的 Toeplitz 矩阵极大左代数族，覆盖了该类矩阵左代数的多种情形。
We study the subfields of quaternion algebras that are quadratic extensions of their center in characteristic 2. We provide examples of the following: two non-isomorphic quaternion algebras that share all their quadratic subfields, two quaternion algebras that share all their inseparable but not all their separable quadratic subfields and two algebras that share all their separable but not all their inseparable quadratic subfields. We also discuss quaternion algebras over global fields and fields of Laurent series over a perfect field of characteristic 2 and show that the quaternion algebras o…
In this paper we provide some applications of the norm form in some quaternion division algebras over rational field and we give some properties of Fibonacci sequence and Fibonacci sequence in connection with quaternion elements. We define a monoid structure over a fnite set on which we will prove that the defined Fibonacci sequence is stationary, we provide some properties of the norm of a rational quaternion algebra, in connection to the famous Lagrange's four-square theorem and its generalizations given by Ramanujan. Moreover, we prove some results regarding the arithmetic of integer quater…
Well-rounded ideal lattices from totally definite quaternion algebras
本文研究了从完全确定四元数代数中得到的完善理想格，证明了其存在性和分类结果，并通过实例展示了研究方法。
In noncommutative and nondivision algebra, left spectrum of matrices are less known and is not easy to handle. Split quaternion algebra is a noncommutative and nondivision algebra. In this paper, by the formulas of solving the equations $ax=b$ and $ax^2+bx+c=0$ over split quaternions, we make an attempt to understand the left spectrum of a split quaternion matrix of order 2.
In the paper I considered linear and antilinear automorphisms of quaternion algebra. I proved the theorem that there is unique expansion of R-linear mapping of quaternion algebra relative to the given set of linear and antilinear automorphisms.
Local superderivation and super-biderivation on generalized quaternion algebra
本文以含单位元交换环上的广义四元数代数为研究对象，将其视作李超代数，探究其上的超双导子与局部超导子，证明了该代数上任意局部超导子均为超导子。
The Donald-Flanigan conjecture asserts that for any finite group and for any field, the corresponding group algebra can be deformed to a separable algebra. The minimal unsolved instance, namely the quaternion group over a field of characteristic 2 was considered as a counterexample. We present here a separable deformation of the quaternion group algebra. In a sense, the conjecture for any finite group is open again.
Maximal prepositive cones on quaternion algebras with involution
本文研究了四元数代数中带反交换的预正锥，证明了对于广泛类别的四元数代数，每个预正锥都是最大的。
We derive the solvability conditions and a formula of a general solution to a Sylvester-type matrix equation over Hamilton quaternions. As an application, we investigate the necessary and sufficient conditions for the solvability of the quaternion matrix equation, which involves $η$-Hermicity. We also provide an algorithm with a numerical example to illustrate the main results of this paper.
The poses of $m$ robotics in $n$ time points may be represented by an $m \times n$ dual quaternion matrix. In this paper, we study the spectral theory of dual quaternion matrices. We introduce right and left eigenvalues for square dual quaternion matrices. If a right eigenvalue is a dual number, then it is also a left eigenvalue. In this case, this dual number is called an eigenvalue of that dual quaternion matrix. We show that the right eigenvalues of a dual quaternion Hermitian matrix are dual numbers. Thus, they are eigenvalues. An $n \times n$ dual quaternion Hermitian matrix is shown to h…
We study possible decompositions of totally decomposable algebras with involution, that is, tensor products of quaternion algebras with involution. In particular, we are interested in decompositions in which one or several factors are the split quaternion algebra $M_2(F)$, endowed with an orthogonal involution. Using the theory of gauges, developed by Tignol-Wadsworth, we construct examples of algebras isomorphic to a tensor product of quaternion algebras with $k$ split factors, endowed with an involution which is totally decomposable, but does not admit any decomposition with $k$ factors $M_2…
In this paper, we considered the theory of quasideterminants and row and column determinants. We considered the application of this theory to the solving of a system of linear equations in quaternion algebra. We established correspondence between row and column determinants and quasideterminants of matrix over quaternion algebra.
Unlike the Hamilton quaternion algebra, the split-quaternions contain nontrivial zero divisors. In general speaking, it is hard to find the solutions of equations in algebras containing zero divisor. In this paper, we manage to derive explicit formulas for computing the roots of $x^{2}+bx+c=0$ in split quaternion algebra.
We study ideal lattices constructed from totally definite quaternion algebras over totally real number fields, and generalize the definition of Arakelov-modular lattices over number fields. In particular, we prove for the case where the totally real number field is $\mathbb{Q}$, that for $\ell$ a prime integer, there always exists a totally definite quaternion over $\mathbb{Q}$ from which an Arakelov-modular lattice of level $\ell$ can be constructed.
Genus of quaternion algebras over formally real fields
本文证明形式实域上四元数可除代数的属有限，并在基本理想第二幂无挠时属平凡，尤其对毕达哥拉斯域成立，方法基于二次型理论。
In this article we give various formulates for compute the number of all coninvolutions over the group of upper triangular matrix with entries into the ring of Gaussian integers module $p$ and the ring of Quaternions integers module $p$, with $p$ an odd prime number.
Let $p$ be a multilinear polynomial in several non-commuting variables with coefficients in an arbitrary field $K$. Kaplansky conjectured that for any $n$, the image of $p$ evaluated on the set $M_n(K)$ of $n$ by $n$ matrices is a vector space. In this paper we settle the analogous conjecture for a quaternion algebra.
Weighted singular value decomposition (WSVD) of a quaternion matrix and with its help determinantal representations of the quaternion weighted Moore-Penrose inverse have been derived recently by the author. In this paper, using these determinantal representations, explicit determinantal representation formulas for the solution of the restricted quaternion matrix equations, ${\bf A}{\bf X}{\bf B}={\bf D}$, and consequently, ${\bf A}{\bf X}={\bf D}$ and ${\bf X}{\bf B}={\bf D}$ are obtained within the framework of the theory of column-row determinants. We consider all possible cases depending on…
In this paper, we investigate and discuss in detail the structures of quaternion tensor SVD, quaternion tensor rank decomposition, and $η$-Hermitian quaternion tensor decomposition with the isomorphic group structures and Einstein product. Then we give the expression of the Moore-Penrose inverse of a quaternion tensor by using the quaternion tensor SVD. Moreover, we consider a generalized Sylvester quaternion tensor equation. We give some necessary and sufficient conditions for the existence of a solution to the generalized Sylvester quaternion tensor equation in terms of the Moore-Penrose inv…
Splitting quaternion algebras defined over a finite field extension
该研究通过分析域上二次型系统的2-扩张迷向性，得到有限域扩张上四元数代数的特定分裂域，并证明次数16的中心单代数可被次数至多2^16的2-扩张分裂。
Within the framework of the theory of the column and row determinants, we obtain determinantal representations of the Drazin inverse for Hermitian matrix over the quaternion skew field. Using the obtained determinantal representations of the Drazin inverse we get explicit representation formulas (analogs of Cramer's rule) for the Drazin inverse solutions of quaternion matrix equations $ {\bf A}{\bf X} = {\bf D}$, $ {\bf X}{\bf B} = {\bf D} $ and ${\bf A} {\bf X} {\bf B} = {\bf D} $, where $ {\bf A}$, ${\bf B}$ are Hermitian.
The product of a complex skew-symmetric matrix and its conjugate transpose is a positive semi-definite Hermitian matrix with nonnegative eigenvalues, with a property that each distinct positive eigenvalue has even multiplicity. This property plays a key role for Professor Loo-Keng Hua to establish the unitary equivalence theorem for complex skew-symmetric matrices. We show that this property is no longer true for quaternion skew-symmetric matrices. This poses a difficulty for finding the canonical form of a quaternion skew-symmetric matrix under unitary equivalence, which may be crucial for th…
The generalized hand-eye calibration matrix equation $AX-YB=C$ over dual quaternions
本文研究双四元数矩阵方程 $AX-YB=C$，给出其可解性条件与通解，并将其用于彩色图像加解密，实验验证了方案的高可行性。
Sylvester-type matrix equations have applications in areas including control theory, neural networks, and image processing. In this paper, we establish the necessary and sufficient conditions for the system of Sylvester-type quaternion matrix equations to be consistent and derive an expression of its general solution (when it is solvable). As an application, we investigate the necessary and sufficient conditions for quaternion matrix equations to be consistent and derive a formula for its general solution involving $η$-Hermicity. As a special case, we also present the necessary and sufficient …
Linear algebra is usually defined over a field such as the reals or complex numbers. It is possible to extend this to skew fields such as the quaternions. However, to the authors' knowledge there is no commonly accepted notation of linear algebra over skew fields. To this end, we discuss ways of notation that account for the non-commutativity of the quaternion multiplication.
Within the framework of the theory of quaternion column-row determinants and using determinantal representations of the Moore-Penrose inverse previously obtained by the author, we get explicit determinantal representation formulas of solutions (analogs of Cramer's rule) to the quaternion two-sided generalized Sylvester matrix equation $ {\bf A}_{1}{\bf X}_{1}{\bf B}_{1}+ {\bf A}_{2}{\bf X}_{2}{\bf B}_{2}={\bf C}$ and its all special cases when its first term or both terms are one-sided. Finally, we derive determinantal representations of two like-Lyapunov equations.
Quasi-determinant and right eigenvalues of dual quaternion matrices
本文研究对偶四元数矩阵的拟行列式与右特征值，证明特征值必为拟特征多项式的根，并给出Hermitian矩阵拟行列式与特征值模平方乘积的等价关系。
In this paper, we construct a simultaneous decomposition of five real quaternion matrices in which three of them have the same column numbers, meanwhile three of them have the same row numbers. Using the simultaneous matrix decomposition, we derive the maximal and minimal ranks of some real quaternion matrices expressions. We also show how to choose the variable real quaternion matrices such that the real quaternion matrix expressions achieve their maximal and minimal ranks. As an application, we give a solvability condition and the general solution to the real quaternion matrix equation $BXD+…
In this paper, we derive some necessary and sufficient solvability conditions for some systems of one sided coupled Sylvester-type real quaternion matrix equations in terms of ranks and generalized inverses of matrices. We also give the expressions of the general solutions to these systems when they are solvable. Moreover, we provide some numerical examples to illustrate our results. The findings of this paper extend some known results in the literature.
We introduce the notion of a confluent Vandermonde matrix with quaternion entries and discuss its connection with Lagrange-Hermite interpolation over quaternions. Further results include the formula for the rank of a confluent Vandermonde matrix, the representation formula for divided differences of quaternion polynomials and their extensions to the formal power series setting.
Within the framework of the theory of the column and row determinants, we obtain explicit representation formulas (analogs of Cramer's rule) for the minimum norm least squares solutions of quaternion matrix equations ${\bf A} {\bf X} = {\bf B}$, $ {\bf X} {\bf A} = {\bf B}$ and ${\bf A} {\bf X} {\bf B} = {\bf D} $.
The fundamental properties of biquaternions (complexified quaternions) are presented including several different representations, some of them new, and definitions of fundamental operations such as the scalar and vector parts, conjugates, semi-norms, polar forms, and inner and outer products. The notation is consistent throughout, even between representations, providing a clear account of the many ways in which the component parts of a biquaternion may be manipulated algebraically.
We present a set of generators for the symplectic group which is different from the well-known set of transvections, from which the chain equivalence for quadratic forms in characteristic 2 is an immediate result. Based on the chain equivalences for quadratic forms, both in characteristic 2 and not 2, we provide chain equivalences for tensor products of quaternion algebras over fields with no nontrivial 3-fold Pfister forms. The chain equivalence for biquaternion algebras in characteristic 2 is also obtained in this process, without any assumption on the base-field.
The Sylvester equation $AX-XB=C$ is considered in the setting of quaternion matrices. Conditions that are necessary and sufficient for the existence of a unique solution are well-known. We study the complementary case where the equation either has infinitely many solutions or does not have solutions at all. Special attention is given to the case where $A$ and $B$ are respectively, lower and upper triangular two-diagonal matrices (in particular, if $A$ and $B$ are Jordan blocks)
In this paper we study special Fibonacci quaternions and special generalized Fibonacci-Lucas quaternions in quaternion algebras over finite fields.
We introduce the notion of maximal orders over quaternion algebras with orthogonal involution and give a classification over local fields, and a partial classification over algebraic number fields.
L.Huang [Linear Algebra Appl. 331 (2001) 21-30] gave a canonical form of a quaternion matrix $A$ with respect to consimilarity transformations $\tilde{S}^{-1}AS$ in which $S$ is a nonsingular quaternion matrix and $\tilde{h}:=a-bi+cj-dk$ for each quaternion $h=a+bi+cj+dk$. We give an analogous canonical form of a quaternion matrix with respect to consimilarity transformations $\hat{S}^{-1}AS$ in which $h\mapsto\hat{h}$ is an arbitrary involutive automorphism of the skew field of quaternions. We apply the obtained canonical form to the quaternion matrix equations $AX-\hat{X}B=C$ and $X-A\hat{X}…
We define algebras of quasi-quaternion type, which are symmetric algebras of tame representation type whose stable module category has certain structure similar to that of the algebras of quaternion type introduced by Erdmann. We observe that symmetric tame algebras that are also 2-CY-tilted are of quasi-quaternion type. We present a combinatorial construction of such algebras by introducing the notion of triangulation quivers. The class of algebras that we get contains Erdmann's algebras of quaternion type on the one hand and the Jacobian algebras of the quivers with potentials associated by …
On the Matrix Form of the Quaternion Fourier Transform and Quaternion Convolution
本研究探讨四元数傅里叶变换与卷积的矩阵形式，阐明其与复数域的关系，并利用理论结果提出限制四元数卷积神经网络Lipschitz常数的方法。
Some comments are made on the matrices which serve as the basis of a quaternionic algebra. We show that these matrices are related with the quaternionic action of the imaginary units from the left and from the right.
This note contains two new observations on the linkage properties of quaternion algebras over fields of characteristic 2: first, that a 3-linked field need not be 4-linked (a case which was left open in previous papers) and that three inseparably linked quaternion algebras are also cyclically linked when the base-field is odd-closed.
Factorization and root-finding for polynomials over division quaternion algebras
针对四元数代数上多项式因式分解与求根研究不足的问题，本文提出了数域上可除四元数代数的多项式因式分解算法，并给出其适配求根的版本，填补了该领域除哈密顿四元数外的研究空白。
Computing square roots in quaternion algebras
本文提出了一种计算四元数代数平方根的算法，适用于特征不为2的全局域。
The determinant of the Laplacian matrix of a quaternion unit gain graph
该研究针对四元数单位增益图的Laplacian矩阵行列式问题，利用新提出的行列非交换行列式给出了其组合描述，并通过数值例子验证了结果。
The group algebras $kQ_{2^n}$ of the generalized quaternion groups $Q_{2^n}$ over fields $k$ which contain $\mathbb{F}_{2^{n-2}}$, are deformed to separable $k((t))$-algebras $[kQ_{2^n}]_t$. The dimensions of the simple components of $\overline{k((t))}\otimes_{k((t))}[kQ_{2^n}]_t$ over the algebraic closure $\overline{k((t))}$, and those of $\mathbb{C} Q_{2^n}$ over $\mathbb {C}$ are the same, yielding strong solutions of the Donald-Flanigan conjecture for the generalized quaternion groups.
Images of Multilinear Polynomials on Generalized Quaternion Algebras
本文推广了多重线性多项式在广义四元数代数上的像的研究，证明其像为向量空间并给出分类，即使代数非可除环。
An elementary proof of the Artin--Springer theorem for generalized quadratic forms over quaternion algebras
本文为四元数除代数上各向异性广义二次型在奇数次域扩张下保持各向异性提供了初等证明。
In this paper we show how generalized quaternions, including 2X2 matrices, can be used to find solutions of a non-commuting equation intimately connected with braid groups. These solutions can then be used to find polynomial invariants of virtual knots and links.
Ideal classes of orders in quaternion algebras
提出一种算法，计算全局域上四元数代数中任意序的所有右理想类代表元，包括不可逆情形，并推广至更一般代数。
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