Study Of The Iterative Method For Some Linear Algebraic Equations Over The Quaternion Field

Quaternion was first discovered by the mathematician Hamilton. Now,quaternion and quaternion matrix are applied to more and more aspects ofscientific research, such as space technology, physics, computer graphics,and robotics industry.Because of the non-commutative multiplication over the quaternionfield, the computer processing is difficult and the research progress ofnumerical calculation is slow. Therefor, the research on quaternion matrix’snumerical algorithm has great practical significance.The chapters1-2in the text are academic background. The chapter1introduces the basic concepts of quaternion and quaternion matrix andsome relevant properties, representing methods and the existed methods tosolve quaternion matrix equations. The chapter2introduces the commonlyused iterative methods over the real number field. The chapters3-4are theauthor’s main research achievements, including:1.Based on the R4block matrix, establish the R4J, R4G-S and R4SORiterative methods and give the necessary and sufficient conditions for theirconvergence.2.Give the definition of R4strictly diagonally dominant matrix over thereal number field, and prove that the strictly diagonally dominant matrixover the quaternion field will be transformed into R4strictly diagonallydominant matrix over the real number field, and the positive definiteself-conjugate quaternion matrices will be transformed into R4block realsymmetric positive definite matrix.3.Based on the strictly diagonally dominant matrix over the quaternion field, the R4J and R4G-S iterative methods can be proved both convergent.Base on the positive definite self-conjugate quaternion matrices, theR4SOR iterative method can be proved convergent too.The numerical results of the5examples designed in this paper aresatisfactory.