Cramer’s Rules for the System of Two-Sided Matrix Equations and of Its Special Cases

Abstract

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, AXB = D, and consequently, AX = D and XB = D are obtained within the framework of the theory of columnrow determinants. We consider all possible cases depending on weighted matrices.

Determinantal Representations of General and (Skew-)Hermitian Solutions to the Generalized Sylvester-Type Quaternion Matrix Equation

“…Then the partial pair solution (43)- (44) to (4), X = ( 푖 푗 ) ∈ H 푛×푛 , Y = ( 푝푔 ) ∈ H 푘×푘 , by the components…”

“…These determinantal representations were used to obtain explicit representation formulas for the minimum norm least squares solutions [38] and weighted Moore-Penrose inverse solutions [39] to some quaternion matrix equations and explicit determinantal representation formulas of both Drazin and W-weighted Drazin inverse solutions to some restricted quaternion matrix equations and quaternion differential matrix equations [40][41][42]. Recently, determinantal representations of solutions to some systems of quaternion matrix equations [43,44] and, in [45], two-sided generalized Sylvester matrix equation 3have been derived by the author as well.…”

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

“…Through the noncommutativity of the quaternion algebra when difficulties arise already in determining the quaternion determinant, the problem of the determinantal representation of generalized inverses only now can be solved due to the theory of column-row determinants introduced in [28,29]. Within the framework of the theory of columnrow determinants, determinantal representations of various kinds of generalized inverses (generalized inverses) solutions of quaternion matrix equations have been derived by the author (see, e.g., [30][31][32][33][34][35][36][37][38][39]) and by other researchers (see, e.g., [40][41][42][43]).…”

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

“…Currently, applying of row-column determinants to determinantal representations of various generalized inverses have been derived by the author (see, e.g. [44][45][46][47][48][49][50][51][52][53][54][55][56][57]) and other researchers (see, e.g. [58][59][60][61]).…”

“…[58][59][60][61]). In particular, determinantal representations of systems like to (1) have been recently explored in [53,55,56,61].…”