Determinantal representations of the W-weighted Drazin inverse over the quaternion skew field

2020

J Math Sci

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 A1X1B1 + A2X2B2 = 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.

Section: Introductionmentioning

confidence: 99%

Determinantal Representations for the Solution of the Generalized Sylvester Quaternion Matrix Equation

2020

J Math Sci

“…e results concerning quaternion matrices have been achieved thanks to the theory of row-column determinants introduced in [28,29]. Within the framework of the theory of row-column determinants, determinantal representations of various kind of generalized inverses, generalized inverse solutions (analogs of Cramer's rule) of quaternion matrix equations have been derived by the author (see, e.g., [30][31][32][33][34][35][36][37][38]) and by other researchers (see, e.g., [39][40][41]).…”

Section: Introductionmentioning

confidence: 99%

Determinantal Representations of the Core Inverse and Its Generalizations with Applications

2019

Journal of Mathematics

Self Cite

In this paper, we give the direct method to find of the core inverse and its generalizations that is based on their determinantal representations. New determinantal representations of the right and left core inverses, the right and left core-EP inverses, and the DMP, MPD, and CMP inverses are derived by using determinantal representations of the Moore-Penrose and Drazin inverses previously obtained by the author. Since the Bott-Duffin inverse has close relation with the core inverse, we give its determinantal representation and its application in finding solutions of the constrained linear equations that is an analog of Cramer’s rule. A numerical example to illustrate the main result is given.

“…Within the framework of the theory of column-row determinants, determinantal representations of various generalized quaternion inverses and generalized inverse solutions to quaternion matrix equations have been derived by the author (see, e.g. [17][18][19][20][21][22][23][24][25][26][27][28]) and by other researchers (see, e.g. [40][41][42][43][44]).…”

Section: Introductionmentioning

confidence: 99%

Determinantal Representations of the Quaternion Core Inverse and Its Generalizations

Adv. Appl. Clifford Algebras

2019

Self Cite

In this paper we extend notions of the core inverse, core EP inverse, DMP inverse, and CMP inverse over the quaternion skew-field H and get their determinantal representations within the framework of the theory of column-row determinants previously introduced by the author. Since the Moore-Penrose inverse and the Drazin inverse are necessary tools to represent these generalized inverses, we use their determinantal representations previously obtained by using row-column determinants. As the special case, we give their determinantal representations for matrices with complex entries as well. A numerical example to illustrate the main result is given.