Search

The solvability of several new constrained quaternion matrix equations is investigated, and their unique solutions are presented in terms of the weighted MPD inverse and weighted DMP inverse of suitable matrices. It is interesting to consider some exceptional cases of these new equations and corresponding solutions. Determinantal representations for the solutions of the equations as mentioned earlier are established as sums of appropriate minors. In order to illustrate the obtained results, a numerical example is shown.

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.

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 WAWX=D, XWBW=D, and W1AW1XW2BW2=D. [ABSTRACT FROM AUTHOR]