Cramer’s rule for some quaternion matrix equations (original) (raw)
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 rule for quaternionic systems of linear equations
Journal of Mathematical Sciences, 2008
New definitions of determinant functionals over the quaternion skew field are given in this paper. The inverse matrix over the quaternion skew field is represented by analogues of the classical adjoint matrix. Cramer's rules for right and left quaternionic systems of linear equations have been obtained.
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.