Study On The Numerical Characteristics Of Quaternion Matrices

Quaternion is an mathematical concept discovered by William Rowan Hamilton(1805-1865) in Ireland in 1843.Specifically, quaternion is another new number system after the complex. The quaternion field is an extension of complex field. It is a relative independent system and has great difference with the complex field because of the non- commutative multiplication of quaternions. Quaternion algebra problems also involve two aspects: Not only does it have abstract theoretical research but also specific practice applications.Quaternion has always been studied by mathematicians and physicists for a century and a half. Especially in the latest 30 years, it is an algebra problem over the quaternion division algebra that has drawn the attention of the researchers of mathematics and physics widely. Many problems of quaternion algebra had been researched, such as the polynomial problem, determinant problem, distribution and estimation of eigenvalues, the system of quaternion matrices equations and so on. It need people to study quaternion algebra problems because of not only the property of non-commutative multiplication of the quaternion, but also its wide-ranging connection with many applied science, such as the quantum physics, geostatic, the figure and pattern recognition and the space telemetry and so forth.In this dissertation, some important algebra features are introduced systematically. The main contents of the dissertation include the following:⑴.For the study of diagonalized quaternion matrices, the necessary and sufficient condition for that is given that the simultaneous diagonalization of quaternion normal matrices and self-conjugate quaternion matrices, by applying some conclusions and methods of simultaneous diagonalization of matrices on the real number field and complex field, meanwhile, we improve it in accordance with the properties of quaternion. Finally, it is discussed that the simultaneous diagonalization of several special matrices.⑵.Based on the concept and similar decomposition of quaternion normal matrices, some properties and judge criterions of quaternion normal matrices are given. Meanwhile, the sufficient conditions for that is given that quaternion normal matrices is unitary similarity to quasi-diagonal matrices by applying the properties of weak direct product. Finally, several eigenvalues inequalities of quaternion normal matrices are obtained. The estimation of the upper and lower bound of the real part and the norm of quaternion normal matrices'right eigenvalues.⑶.Some properties and judge criterions of skew self-conjugate quaternion matrices are gained by applying its concept. Meanwhile, several theorems about real expression, similar decomposition and eigenvalues of skew self-conjugate quaternion matrices are obtained.⑷.By applying the properties of several inequalities of the trace on the real number field and complex field,at the same time, we improve them in accordance with the properties of quaternion and obtain several inequalities of the trace of quaternion matrices.