Study On The Distribution And Estimate Of Algebra Eigenvalues And Singular Values Decompose Over Real Quaternion Field

Algebra and numerical algebra are two important parts of computational mathematics. Quaternion algebra is the extension of algebra over complex field. However, because of the non-commutative multiplication of quaternion, it leads to two cases: There are not only some connections, but also vital differences between quaternion algebra and complex algebra, and it have formed an independent relatively algebra system. Quaternion algebra problems also involve two aspects: Not only does it have abstract theoretical research but also specific practice applications.In recent years, the algebra problem over quaternion division algebra has drawn the attention of researchers of mathematics and physics. Many problems of quaternion division algebra have been studied, such as polynomial, determinant, eigenvalue, and system of quaternion matrices equations and so on. It is not easy to study quaternion algebra problems because of the non-commutative multiplication of quaternions. However, quaternion algebra theory is getting more and more important. In many fields of applied science, such as physics, figure and pattern recognition, spacecraft attitude control, 3-D animation and so forth, people start making use of quaternion algebra theory to solve some actual problems. Therefore, it is necessary make a further study about quaternion algebra theory.Theoretically, many algebra problems on quaternion filed need studying and solving, such as distribution and estimation of real quaternion matrix eigenvalues, quaternion polynomial, and singular value decomposition of quaternion matrix, solution to the system of quaternion equations, canonical form and positive definite problems of quaternion matrix and so on.In this dissertation, some important algebra features are introduced systematically. The concepts of norm and generalized spherical neighborhood of quaternion are established, the location of quaternion matrix eigenvalues, connectivity of spherical neighborhood of eigenvalues, the smallest inclusion theorem and upper bound and lower bound estimation theorems for the moment, the real and imaginary part of the real quaternion matrices eigenvalues are studied. Furthermore, the location and estimation of quaternion matrices left and right eigenvalues are studied and some theorems are obtained. Some inequalities about eigenvalues of sum, difference and tensor product of matrixes on quaternion division algebra are also obtained. Based on the well-known Gerschgorin Theorem, the form of Cassini Theorem is solved on quaternion division algebra. Besides, theorems about singular value decomposition of Kronecker and Hadamard product of quaternion matrix are obtained.In this dissertation, the positive definite matrix is introduced in previous literatures which are divided into four types, namely,â… ,â…¡,â…¢andâ…£positive definite matrices, of which the sets are denoted by S_â… (M), S_â…¡(M), S_â…¢(M) and S_â…£(M) respectively and some judgment theorems about positive definite matrix are obtained. Finally, for the sake of state the variety of quaternion algebra problems, paper introduced some other conclusions about diagonalizable of normal matrix and location of algebra eigenvalues, such as the new Gerschgorin theorem, Ostrowski theorem and Brauer theorem, they are the generalization of related theorem on complex field.