Dual Self-conjugate Solution Of The Quaternion Matrix Equation And Inverse Quadratic Eigenvalue Problems

Some structural solutions and inverse eigenvalue problems of matrix equations are hot topics in the field of matrix calculation.However,people mainly focus on the study of complex matrices.Little research has been done on the inverse problem between the structural solution and the quadratic eigenvalue of quaternion equations.In this dissertation,we study the double self-conjugate matrix solutions and the optimal approximation solution of the quaternion matrix equations.The quadratic eigenvalue inverse problem of Hermitian R-(skew)symmetric matrix is discussed.The details are as follows:1.The background of the inverse problem of matrix equations and quadratic eigenvalues is summarized.This paper points out the research status and progress at home and abroad,and gives some preliminary knowledge about the definition and nature.2.The double self-conjugated(positive definite)solution and its inverse problem solution of continuous Lyapunov equation AX+XA~*=B are studied.In the double self conjugate solution set of the equation,the optimal approximation solution with minimal Frobenius norm is found.Numerical examples are used to verify the feasibility of the proposed method.3.We study the least squares double conjugate solution and its optimal approximation problem of quaternion matrix equations AX=B,XC=D.We Mainly used the structural properties of double self-conjugated matrices,the singular value decomposition of matrix pairs and other techniques,the solution expression of this problem is obtained.The correctness and feasibility of the proposed method are tested by numerical example.4.The inverse problem of quadratic eigenvalues of Hermitian R-(skew)symmetric matrix in complex domain is discussed.Based on the structural characteristics of Hermitian R-(skew)symmetric Matrix,the original problem is transformed into equations solving problem.Using the symmetry of the matrix and the Kronecker product,The general expression of the solution of the original problem is obtained.