Structure-preserving Algorithms For Generalized Schur Decomposition Of Quaternion Matrix Pencil With Applications

Generalized Schur decomposition is an important part of matrix theory research and a commonly used method for solving generalized eigenvalue problems.In recent years,quaternion models have become increasingly widely used in fields such as color images,pattern recognition,and signal processing due to their unique construction.This thesis is based on the idea of quaternion structure-preserving method and investigates the generalized Schur decomposition method of quaternion models,which is used to solve the generalized eigenvalue problem of quaternion regular matrix pencil and color face recognition.This thesis mainly proposes the generalized Schur decomposition QZ iteration algorithm of quaternion matrix pencil and generalized Schur decomposition cross iteration algorithm of quaternion matrix pencil.In order to further apply them in practice,the thesis proposes a quaternion regular matrix pencil generalized eigenvalue algorithm and a quaternion bilateral two-dimensional linear discriminant analysis model based on the generalized Schur decomposition algorithm.Firstly,based on the connections between different channels of colored facial images,and taking into account the information that cannot be ignored between rows and columns of the image matrix,a quaternion bilateral two-dimensional linear discriminant analysis model is proposed to handle the problem of color facial image recognition.Experiments have shown that this algorithm has superior recognition rate and CPU running time compared to other LDA class models.Secondly,based on the quaternion structure-preserving algorithm and generalized Schur decomposition algorithm of the matrix pencil,a generalized Schur decomposition algorithm of quaternion matrix pencil is proposed.The entire process of quaternion matrix pencil double step displacement QZ iteration is introduced in detail by using matrix form,and an efficient and stable generalized Schur decomposition structure-preserving algorithm of quaternion matrix pencil is designed.Thirdly,a generalized Schur decomposition cross iteration algorithm for quaternion matrix pencil is proposed.Based on the upper Hessenberg-upper triangle reduction theory,the process of upper Hessenberg-upper triangle cross iteration on quaternion matrix pencils is demonstrated.In addition,in order to analyze more information about the generalized eigenvalues of quaternion matrix pencil,this thesis also proposes a quaternion regular matrix pencil generalized eigenvalue decomposition algorithm based on the generalized Schur decomposition algorithm.Finally,a series of numerical experiments were designed to validate the effectiveness of the model and algorithm,including experiments on solving the generalized eigenvalues of regular matrix pencil and experiments on color facial image recognition,which demonstrated the practical applicability of the algorithm.