Spectral Analysis Of Large Dimensional Quaternion Random Matrices With Applications

Random Matrix Theory is a popular subject with many applications in practice,e-specially in the area of quantum physics and High-dimensional statistics.As one of the classical models in random matrix theory,quaternion self-dual Hermitian random matri-ces are very meaningful in quantum physics.At the same time,In the sense of algebraic,the study of quaternions as a kind of hyper-complex is a critical step of solving problems within Clifford algebra.In fact,according to the Frobenius theorem,the real number field R,complex number field C and quaternion field Q are the only three finite-dimensional real algebras satisfy that each nonzero element has an inverse.Combining with the theo-rem,we know that every finite-dimensional semi-simple algebra can be written as a direct sum of matrix algebras with real,complex or quaternion entries.In particular,this is true for the Clifford algebra(Geometric algebra).Thus,if certain properties is true for the re-al,complex and quaternion matrix,so do these properties for all the matrix with Clifford algebra entries.This important properties makes it important to make extensions of some properties form real and complex field to quaternion field.What' s more,quaternions find more and more applications recently.It has many advantages in soling problems in the areas such as signal processing,color image processing and so on.In this paper,we will investigate the spectral analysis of large dimensional quater-nion random matrices.Instead of GSE,we will consider the spectral properties of large dimensional quaternion random matrices without Gaussian conditions,which we called"universality".Under some moment conditions,we give some basic spectral properties of quaternion random matrix with the dimension tends to infinity.To specify,the first chapter is the introduction of this thesis which give a brief introduction of Random Ma-trix Theory as well as some basic knowledge of quaternion algebra and some applications of quaternion and quaternion matrix.The second chapter consider the limiting spectral distribution of large dimensional random quaternion self-dual Hermitian matrix.Under a Lindeberg type condition,we show that the empirical spectral distribution will converge to the semicircle law.Correspondingly,we also consider the limiting spectral distribution of generalized quaternion sample covariance matrix in the third chapter.This result can be applied to the quaternion time series and quaternion linear process models thus of high practical values.The fourth chapter is to study the limit properties of extreme eigenvalues.We give the necessary and sufficient conditions for the almost surely convergence of the extreme eigenvalues of large dimensional quaternion self-dual Hermitian random matrix.At last,the final chapter gives a convergence rate of the empirical spectral distribution.