Structure, properties and applications of some simultaneous decompositions for quaternion matrices involving \(\phi\)-skew-Hermicity
type: Item
instance of: scholarly article
title: Structure, properties and applications of some simultaneous decompositions for quaternion matrices involving \(\phi\)-skew-Hermicity
zbMATH Open document ID: 1409.15017
author: Zhuo-Heng He
published in: Advances in Applied Clifford Algebras
publication date: 2019-02-22T00:00:00Z
review text: Let $\mathbb{H}^{m\times n}$ be the set of all $m \times n$ matrices over the real quaternion algebra \[ \mathbb{H}=\{ a_0 +a_1\mathbf{i}+a_2\mathbf{j}+a_3\mathbf{k}\mid \mathbf{i}^2 =\mathbf{j}^2 =\mathbf{k}^2 =\mathbf{ijk}=-1,\; a_0,a_1,a_2,a_3 \in \mathbb{R}. \] For $A \in \mathbb{H}^{m\times n}$, we denote by $A_{\phi}$ the $n\times m$ matrix obtained by applying $\phi$ entrywise to the transposed matrix $A^t$, where $\phi$ is a nonstandard involution of $\mathbb{H}$. The matrix $A \in \mathbb{H}^{n\times n}$ is said to be $\phi$-skew-Hermitian if $A = -A_{\phi}$. \par The authors investigate the structure, the properties, and some applications of some simultaneous decomposition for six quaternion matrices: \[ \begin{pmatrix} A&B&C&&\\ &&D&E&F \end{pmatrix} \] where $A,B,C, D, E, F\in \mathbb{H}^{n\times n}$ are $\phi$-skew-Hermitian matrices. \par The paper is well written and contains interesting results.
reviewed by: Yansheng Wu
Mathematics Subject Classification ID: 15B33
Mathematics Subject Classification ID: 15A23
Mathematics Subject Classification ID: 15A24
Mathematics Subject Classification ID: 15B48
Mathematics Subject Classification ID: 15B57
zbMATH DE Number: 7027674
zbMATH Keywords: quaternion
zbMATH Keywords: matrix decomposition
zbMATH Keywords: matrix equation
zbMATH Keywords: $\beta(\phi)$-signature
zbMATH Keywords: $\phi$-skew-Hermitian solution
MaRDI profile type: Publication
full work available at URL: s00006-018-0921-4
OpenAlex ID: W2900590900
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DOI: 10.1007/S00006-018-0921-4
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description: scientific article; zbMATH DE number 7027674