Research On Some Problems Of Quaternion Matrix And Framework

Quaternions have been widely used in many aspects such as:quantum mechan-ics,computer science,programming video games,the control of spacecrafts and so on.Linear algebra over quaternions is not far removed from classical linear algebra.Because of the noncommutative multiplication of quaternions,the left and right eigenvalues of quaternionic matrices must be considered respectively in the process of theoretical research.And their numbers and numberical values have big differ-ences in general.Resently,Gerschgorin types for all the left eigenvalues and part of right eigenvalues of quaternionic matrices have been proposed by some scholars.The second chapter of this paper,we achieve the estimation of all the right eigenvalues of quaternionic matrix by adding some constraints,on the basis of the conclusions that have been got.According to the theorem,we can get the range of all the right eigenvalues,as long as the matrix meets the conditions of it.Our conclusions are more valuble than the previous achievement.When we solve the system of linear equations,the Jacobi iterative method and Gauss-Seidel iterative method are convergent if the coefficient matrix is strictly and diagonally dominant.So the strictly diagonally dominant matrix has very good property in the research of the convergence of iterative method.The third chapter of this paper,we define the strictly diagonally dominant quaternionic matrix and the infinite norm of quaternionic matrix,then discuss the convergent condition of Jacobi iterative method,Jacobi iterative method of extrapolation,Jacobi iterative method of parallel double parameters with the infinite norm.At last combining the conclusion what the second chapter has got,we give some numerical examples to show their validity.Frames have turned out to be an essential tool for many applications.Their main advantage lies in the fact that a frame can be designed to be redundant while still providing a reconstruction formula.And equiangular tight frames play an important role in several areas of mathematics,ranging from signal processing to quantum computing.The forth part of this paper,we give the definition of the quaternionic frames,and demonstrate the existence of quaternionic equiangular tight frames.Then some numerical examples are given to show the existence of frames.