Cramer’s rules for the system of quaternion matrix equations with η-Hermicity

• Main Author: Kyrchei Ivan I.
• Format: Article
• Language: English
• Published: EDP Sciences 2019-01-01
• Series: 4 open
• Subjects: Generalized inverse; Noncommutative determinant; Quaternion matrix; System of matrix equations; Cramer rule; η-Hermicity
• Online Access: https://www.4open-sciences.org/articles/XXXXX/full_html/2019/01/fopen190008/fopen190008.html

The system of two-sided quaternion matrix equations with η-Hermicity, A1XA1η* = C1 A 1 X A 1 η * = C 1 $ {\mathbf{A}}_1\mathbf{X}{\mathbf{A}}_1^{\eta \mathrm{*}}={\mathbf{C}}_1$ , A2XA2η* = C2 A 2 X A 2 η * = C 2 $ {\mathbf{A}}_2\mathbf{X}{\mathbf{A}}_2^{\eta \mathrm{*}}={\mathbf{C}}_2$ is considered in the paper. Using noncommutative row-column determinants previously introduced by the author, determinantal representations (analogs of Cramer’s rule) of a general solution to the system are obtained. As special cases, Cramer’s rules for an η-Hermitian solution when C1 = Cη*1 C 1 = C 1 η * $ {\mathbf{C}}_1={\mathbf{C}}_1^{\eta \mathrm{*}}$ and C2 = Cη*2 C 2 = C 2 η * $ {\mathbf{C}}_2={\mathbf{C}}_2^{\eta \mathrm{*}}$ and for an η-skew-Hermitian solution when C1 = −Cη*1 C 1 = - C 1 η * $ {\mathbf{C}}_1=-{\mathbf{C}}_1^{\eta \mathrm{*}}$ and C2 = −Cη*2 C 2 = - C 2 η * $ {\mathbf{C}}_2=-{\mathbf{C}}_2^{\eta \mathrm{*}}$ are also explored.