Cramer’s rule for some quaternion matrix equations (original) (raw)
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Cramer's rules for the solution to the two-sided restricted quaternion matrix equation
2017
Weighted singular value decomposition (WSVD) of a quaternion matrix and with its help determinantal representations of the quaternion weighted Moore-Penrose inverse have been derived recently by the author. In this paper, using these determinantal representations, explicit determinantal representation formulas for the solution of the restricted quaternion matrix equations, bfAbfXbfB=bfD{\bf A}{\bf X}{\bf B}={\bf D}bfAbfXbfB=bfD, and consequently, bfAbfX=bfD{\bf A}{\bf X}={\bf D}bfAbfX=bfD and bfXbfB=bfD{\bf X}{\bf B}={\bf D}bfXbfB=bfD are obtained within the framework of the theory of column-row determinants. We consider all possible cases depending on weighted matrices.
Cramer’s rules for the system of quaternion matrix equations with η-Hermicity
4open
The system of two-sided quaternion matrix equations with η-Hermicity, A1XA1η* = C1 , A2XA2η* = C2 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 and C2 = Cη*2 and for an η-skew-Hermitian solution when C1 = −Cη*1 and C2 = −Cη*2 are also explored.
Cramer’s Rules for the System of Two-Sided Matrix Equations and of Its Special Cases
Matrix Theory - Applications and Theorems, 2018
Within the framework of the theory of row-column determinants previously introduced by the author, we get determinantal representations (analogs of Cramer's rule) of a partial solution to the system of two-sided quaternion matrix equations A 1 XB 1 =C 1 , A 2 XB 2 =C 2. We also give Cramer's rules for its special cases when the first equation be one-sided. Namely, we consider the two systems with the first equation A 1 X=C 1 and XB 1 =C 1 , respectively, and with an unchanging second equation. Cramer's rules for special cases when two equations are one-sided, namely the system of the equations A 1 X=C 1 , XB 2 =C 2 , and the system of the equations A 1 X=C 1 , A 2 X=C 2 are studied as well. Since the Moore-Penrose inverse is a necessary tool to solve matrix equations, we use its determinantal representations previously obtained by the author in terms of row-column determinants as well.
The General Solution of Quaternion Matrix Equation Having η-Skew-Hermicity and Its Cramer’s Rule
Mathematical Problems in Engineering
We determine some necessary and sufficient conditions for the existence of the η-skew-Hermitian solution to the following system AX-(AX)η⁎+BYBη⁎+CZCη⁎=D,Y=-Yη⁎,Z=-Zη⁎ over the quaternion skew field and provide an explicit expression of its general solution. Within the framework of the theory of quaternion row-column noncommutative determinants, we derive its explicit determinantal representation formulas that are an analog of Cramer’s rule. A numerical example is also provided to establish the main result.
Journal of Mathematics
Within the framework of the theory of quaternion row-column determinants previously introduced by the author, we derive determinantal representations (analogs of Cramer’s rule) of solutions and Hermitian solutions to the system of two-sided quaternion matrix equations A1XA1⁎=C1 and A2XA2⁎=C2. Since the Moore-Penrose inverse is a necessary tool to solve matrix equations, we use determinantal representations of the Moore-Penrose inverse previously obtained by the theory of row-column determinants.
arXiv: Rings and Algebras, 2018
Within the framework of the theory of quaternion column-row determinants and using determinantal representations of the Moore-Penrose inverse previously obtained by the author, we get explicit determinantal representation formulas of solutions (analogs of Cramer's rule) to the quaternion two-sided generalized Sylvester matrix equation $ {\bf A}_{1}{\bf X}_{1}{\bf B}_{1}+ {\bf A}_{2}{\bf X}_{2}{\bf B}_{2}={\bf C}$ and its all special cases when its first term or both terms are one-sided. Finally, we derive determinantal representations of two like-Lyapunov equations.
2016
Weighted singular value decomposition (WSVD) and a representation of the weighted Moore-Penrose inverse of a quaternion matrix by WSVD have been derived. Using this representation, limit and determinantal representations of the weighted Moore-Penrose inverse of a quaternion matrix have been obtained within the framework of the theory of the noncommutative column-row determinants. By using the obtained analogs of the adjoint matrix, we get the Cramer rules for the weighted Moore-Penrose solutions of left and right systems of quaternion linear equations.