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

2019

Journal of Mathematics

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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.

“…, then it follows (40). e determinantal representation (41) can be obtained similarly by integrating (9) for the determinantal representation of (A k+1 ) † in (37).…”

Section: Lemmamentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Determinantal Representations of the Core Inverse and Its Generalizations with Applications

2019

Journal of Mathematics

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“…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]).…”

Section: Introductionmentioning

confidence: 99%

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

2019

4open

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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.

“…Our proposed Cramer's rule is based on the theory of rowcolumn noncommutative determinants introduced in [34], by using determinantal representations of the Moore-Penrose inverse matrix [35]. Within the framework of the theory of noncommutative row-column determinants, determinantal representations of various generalized quaternion inverses and generalized inverse solutions to quaternion matrix equations have been derived by one of the authors (see, e.g., [36][37][38][39][40][41][42][43][44][45][46]) and by other researchers (see, e.g., [47][48][49][50][51]). Observe that the system (3) is a particular case of our system (5).…”

Section: Introductionmentioning

confidence: 99%

The General Solution of Quaternion Matrix Equation Having η‐Skew‐Hermicity and Its Cramer’s Rule

Rehman

Mathematical Problems in Engineering

Ali

et al. 2019

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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.