Ivan Kyrchei
New definitions of determinant functionals over the quaternion skew field are given in this paper. The inverse matrix over the quaternion skew field is represented by analogues of the classical adjoint matrix. Cramer rule for right and left quaternionic systems of linear equations have been obtained.
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Weighted singular value decomposition (WSVD) and a representation of the weighted Moore-Penrose inverse of a quaternion matrix by WSVD have been derived. Using this representation, limit and determinantal representations of the weighted Moore-Penrose inverse of a quaternion matrix have been obtained within the framework of the theory of the noncommutative column-row determinants. By using the obtained analogs of the adjoint matrix, we get the Cramer rules for the weighted Moore-Penrose solutions of left and right systems of quaternion linear equations.
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.
Determinantal representation of the Moore-Penrose inverse over the quaternion skew field is obtained within the framework of a theory of the column and row determinants. Using the obtained analogs of the adjoint matrix, we get the Cramer rules for the least squares solution of left and right systems of quaternionic linear equations.
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…
The matrix equation $AX-XB=C$ has a solution if and only if the matrices [A&C\\0&B] and [A &0\\0 & B] are similar. This criterion was proved over a field by W.E. Roth (1952) and over the skew field of quaternions by Huang Liping (1996). H.K. Wimmer (1988) obtained an analogous criterion for the matrix equation $X-AXB=C$ over a field. We extend these criteria to the matrix equations $AX-\widehat XB=C$ and $X-A\widehat XB=C$ over the skew field of quaternions with a fixed involutive automorphism $q\mapsto \hat q$.
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…
We introduce a class of rings using which we define the concept of skew regularity for quaternion-valued functions over quaternions. It is shown that the notion of skew regularity coincides with the concept of slice regularity over symmetric slice domains. Known results regarding slice-regular functions over symmetric slice domains are generalized to skew-regular functions over general symmetric domains. Furthermore, we present new results concerning slice-regular functions.
Diagonalization of skew-Hermitian dual quaternion matrices and $ϕ$-Hermitian dual quaternion matrices
本文研究斜厄米对偶四元数矩阵的酉等价对角化，以及单位四元数$\phi$对应的$\phi$-厄米对偶四元数矩阵的酉$\phi$-相合对角化，推广了相关矩阵分解结论。
Within the framework of the theory of the column and row determinants, we obtain new determinantal representations of the W-weighted Drazin inverse over the quaternion skew field. We give determinantal representations of the W-weighted Drazin inverse by using previously introduced determinantal representations of the Drazin inverse, the Moore-Penrose inverse, and the limit representations of the W-weighted Drazin inverse in some special case.
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.
A basic theory on the first order right and left linear quaternion differential systems (LQDS) is given systematic in this paper. To proceed the theory of LQDS we adopt the theory of column-row determinants recently introduced by the author. In this paper, the algebraic structure of their general solutions are established. Determinantal representations of solutions of systems with constant coefficient matrices and sources vectors are obtained in both cases when coefficient matrices are invertible and singular. In the last case, we use determinantal representations of the quaternion Drazin inve…
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}…
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} $.
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.
By using determinantal representations of the W-weighted Drazin inverse previously obtained by the author within the framework of the theory of the column-row determinants, we get explicit formulas for determinantal representations of the W-weighted Drazin inverse solutions (analogs of Cramer's rule) of the quaternion matrix equations $ {\bf W}{\bf A}{\bf W}{\bf X}={\bf D}$, $ {\bf X}{\bf W}{\bf A}{\bf W}={\bf D} $, and ${\bf W}_{1}{\bf A}{\bf W}_{1}{\bf X}{\bf W}_{2}{\bf B}{\bf W}_{2}={\bf D} $.
In this paper we extend notions of the core inverse, core EP inverse, DMP inverse, and CMP inverse over the quaternion skew-field ${\mathbb{H}}$ and get their determinantal representations within the framework of the theory of column-row determinants previously introduced by the author. Since the Moore-Penrose inverse and the Drazin inverse are necessary tools to represent these generalized inverses, we use their determinantal representations previously obtained by using row-column determinants. As the special case, we give their determinantal representations for matrices with complex entries …
In this paper, we extend notions of the weighted core-EP right and left inverses, the weighted DMP and MPD inverses, and the CMP inverse to matrices over the quaternion skew field H that have some features in comparison to these inverses over the complex field. We give the direct methods of their computing, namely, their determinantal representations by using noncommutative column and row determinants previously introduced by the author. As the special cases, by using the usual determinant, we give their determinantal representations for matrices with complex entries as well. A numerical examp…
The $ϕ$-conjugation of quaternionic matrices and generalized Autonne-Takagi factorization
本文针对模为1的四元数$\phi$，研究四元数矩阵的$\phi$-共轭相关性质，并将Autonne-Takagi分解推广至所有单位四元数，肯定回答了Horn与Zhang提出的问题。
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.
Hensel's lemma for the norm principle for spinor groups
本文在规范原理成立的条件下，证明了旋量群的赫尔曼引理在四元数代数上的有效性。
$\boldsymbol{i}$-conjugate for quaternionic matrices and related properties
本文受复矩阵酉等价于实矩阵的启发，将其推广至四元数，提出$\boldsymbol{i}$-共轭概念，给出四元数矩阵酉等价于复矩阵的充要条件，并研究相关性质及新型极分解。
As is well known, the common elementary functions defined over the real numbers can be generalized to act not only over the complex number field but also over the skew (non-commuting) field of the quaternions. In this paper, we detail a number of elementary functions extended to act over the skew field of Clifford multivectors, in both two and three dimensions. Complex numbers, quaternions and Cartesian vectors can be described by the various components within a Clifford multivector and from our results we are able to demonstrate new inter-relationships between these algebraic systems. One key…
In this paper we investigate the Brolin's theorem over $\mathbb{H}$, the skew field of quaternions. Moreover, considering a quaternionic polynomial $p$ with real coefficients, we focus on the properties of its equilibrium measure, among the others, the mixing property and the Lyapunov exponents of the measure. We prove a central limit theorem and we compute the topological entropy and measurable entropy with respect to the quaternionic equilibrium measure. We prove that they are equal considering both a quaternionic polynomial with real coefficients and a polynomial with coefficients in a slic…
We consider polynomials of bi-degree $(n,1)$ over the skew field of quaternions where the indeterminates commute with each other and with all coefficients. Polynomials of this type do not generally admit factorizations. We recall a necessary and sufficient condition for existence of a factorization with univariate linear factors that has originally been stated by Skopenkov and Krasauskas. Such a factorization is, in general, non-unique by known factorization results for univariate quaternionic polynomials. We unveil existence of bivariate polynomials with non-unique factorizations that cannot …
We generalize linear superalgebra to higher gradings and commutation factors, given by arbitrary abelian groups and bicharacters. Our central tool is an extension, to monoidal categories of modules, of the Nekludova-Scheunert faithful functor between the categories of graded-commutative and supercommutative algebras. As a result we generalize (super-)trace, determinant and Berezinian to graded matrices over graded-commutative algebras. For instance, on homogeneous quaternionic matrices, we obtain a lift of the Dieudonné determinant to the skew-field of quaternions.
Orthogonal involutions over fields with I^3=0
本研究针对特征不为2且I^3F=0的基域F，推导了带典范对合的四元数代数上斜埃尔米特型的u不变量的上界，上界由基域的u不变量表出。
We show that the spectral theorem -- which we understand to be a statement that every self-adjoint matrix admits a certain type of canonical form under unitary similarity -- admits analogues over other $*$-algebras distinct from the complex numbers. If these $*$-algebras contain nilpotents, then it is shown that there is a consistent way in which many classic matrix decompositions -- such as the Singular Value Decomposition, the Takagi decomposition, the skew-Takagi decomposition, and the Jordan decomposition, among others -- are immediate consequences of these. If producing the relevant canon…
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