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On the solutions of some linear complex quaternionic equations
1Department of Mathematics, Faculty of Art and Science, Mustafa Kemal University, Tayfur Sökmen Campus, 31100 Hatay, Turkey.
Thescientificworldjournal
|August 8, 2014
Summary
This study investigates complex quaternionic equations, known as generalized Sylvester-quaternion equations. Researchers determined the conditions for solving these equations and derived general solutions using real matrix representations.
Area of Science:
- Mathematics
- Linear Algebra
- Quaternion Theory
Background:
- Sylvester equations are fundamental in control theory and systems analysis.
- Complex quaternions extend quaternion algebra with additional applications.
- Generalized Sylvester-quaternion equations present a more complex mathematical challenge.
Purpose of the Study:
- To analyze and solve a specific class of complex quaternionic equations: AX - XB = C.
- To establish necessary and sufficient conditions for the solvability of these equations.
- To derive general expressions for the solutions of the investigated equations.
Main Methods:
- Utilizing real matrix representations of complex quaternions.
- Applying techniques from linear algebra to transform the quaternion equations into matrix form.
- Deriving solvability conditions and solution structures based on the matrix representations.
Main Results:
- The necessary and sufficient conditions for the existence of solutions to generalized Sylvester-quaternion equations are established.
- General formulas for the solutions of these equations are derived.
- The study demonstrates the effectiveness of real matrix representations for solving complex quaternionic systems.
Conclusions:
- The research provides a comprehensive method for solving generalized Sylvester-quaternion equations.
- The findings contribute to the understanding and application of quaternion algebra in solving matrix equations.
- This work offers a valuable tool for researchers in algebra and related fields.
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