Cramer’s rules for the system of quaternion matrix equations with η-Hermicity (original) (raw)

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.

Explicit formulas and determinantal representation for η-skew-Hermitian solution to a system of quaternion matrix equations

Filomat, 2020

Some necessary and sufficient conditions for the existence of the η-skew-Hermitian solution quaternion matrix equations the system of matrix equations with η-skew-hermicity, A1X = C1, XB1 = C2, A2Y = C3, YB2 = C4, X = −Xη∗, Y = −Yη∗, A3XA η∗ 3 + B3YB η∗ 3 = C5, are established in this paper by using rank equalities of the coefficient matrices. The general solutions to the system and its special cases are provided when they are consistent. Within the framework of the theory of noncommutative row-column determinants, we also give determinantal representation formulas of finding their exact solutions that are analogs of Cramer’s rule. A numerical example is also given to demonstrate the main results.

Determinantal Representations of Solutions and Hermitian Solutions to Some System of Two-Sided Quaternion Matrix Equations

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.

Cramer’s rule for some quaternion matrix equations

Applied Mathematics and Computation, 2010

Cramer's rules for some left, right and two-sided quaternion matrix equations are obtained within the framework of the theory of the column and row determinants.

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.