Algorithms For Joint Diagonalization Of Quaternion Covariance Matrices

With the development of blind source separation technology and the advantages of quaternion in multi-dimension data representation and application,the separation processing algorithm of quaternion signal has become a research hotspot in the field of mathematics and signal processing.This paper mainly studies on some problems of algorithms of joint diagonalization of quaternion covariance matrix and complementary covariance matrix.On the basis of maintaining the real representation structure of matrices,we put forward effective algorithms for the corresponding problems and verify the effectiveness of the algorithms through numerical experiments.In chapter 1,we introduce the research background,research status and main research content.And we introduce the development background of blind source signals and the development status of the joint diagonalization algorithms in detail,we also provide a practical background for the research work in the following.In chapter 2,we mainly present the basic properties of real representation and corresponding relationship with quaternion matrix.Based on the real representation of quaternion matrix,we introduce the structure-preserving feature decomposition algorithm,singular Value decomposition algorithm, decomposition algorithm and random singular value decomposition algorithm of quaternion matrix which are basis for the structurepreserving joint diagonalization algorithm in the following chapters.In chapter 3,we investigate the research background of joint diagonalization of two quaternion Hermitian matrices and propose the theoretical results and analysis of joint diagonalization algorithm.We bring forward quaternion Hermitian matrix joint diagnolization algorithm and structure-preserving joint diagonalization algorithm on the basis of theoretical conditions.In chapter 4,we study the joint diagonalization problem of the quaternion covariance matrix and the complementary covariance matrix.The correlation properties of the covariance matrix and the complementary covariance matrix are briefly explained.We give the conditions for joint diagonalization of quaternion covariance matrix and quaternion complementary covariance matrix,and propose a structure-preserving joint diagonalization algorithm.In chapter 5,we choose different matrices to experiment and analyze the advantages and disadvantages of the algorithms proposed in chapter 2 and chapter 4.By selecting different matrices for experiments and combining the experimental results,the necessary algorithm analysis and explanation are carried out.