Structure, properties and applications of some simultaneous decompositions for quaternion matrices involving \(\phi\)-skew-Hermicity
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.
scientific article; zbMATH DE number 7027674
scientific article