Aleks Kleyn, Ivan Kyrchei
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.
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We compute quaisideterminants and determinants of quaternionic matrices
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.
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…
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 …
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.
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+…
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 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} $.
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.
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…
Let $\mathbb{H}$ be the real quaternion algebra and $\mathbb{H}^{m\times n}$ denote the set of all $m\times n$ matrices over $\mathbb{H}$. Let $\mathbf{i},\mathbf{j},\mathbf{k}$ be the imaginary quaternion units. For $η\in\{\mathbf{i},\mathbf{j},\mathbf{k}\}$, a square real quaternion matrix $A$ is said to be $η$-Hermitian if $A^{η*}=A$ where $A^{η*}=-ηA^{\ast}η$, and $A^{\ast}$ stands for the conjugate transpose of $A$. In this paper, we construct a simultaneous decomposition of four real quaternion matrices with the same row number $(A,B,C,D),$ where $A=A^{η*}\in \mathbb{H}^{m\times m}, B\in…
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。
Associated to any complex simple Lie algebra is a non-reductive complex Lie algebra which we call the intermediate Lie algebra. We propose that these algebras can be included in both the magic square and the magic triangle to give an additional row and column. The extra row and column in the magic square corresponds to the sextonions. This is a six dimensional subalgebra of the split octonions which contains the split quaternions.
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