Iterative Algorithms For Quaternion Matrix Equations With Structural Constraints

The constraint matrix equation problem is to find the solution of the equation in the set of matrices satisfying certain constraints.Different constraints and equations will produce new research problems.Constrained matrix equations is the research hotspots of current numerical algebra and have important applications in many fields such as structural design,parameter identification,automatic control,vibration theory and nonlinear programming.Based on the theory of conjugate gradient and using Matlab as a tool,this paper establishes an iterative algorithm for the structure of three kinds of quaternion matrix equations.The specific contents are as follows:1.The central self-conjugate solution of the matrix equation AX=B is discussed in the complex domain and the quaternion.Using the structural features of the central self-conjugate matrix,transform the structural constraint equation into an unconstrained equation,and establish two algorithms of parameter iteration and conjugate gradient iteration of the equation,and analyzes the convergence of the algorithm.2.The cyclic solution of the quaternion matrix equation AXB +CXD=E is discussed.According to the special structure of the quaternion cyclic matrix,the real decomposition of the quaternion matrix and the Kronecker product is used to transform the constrained quaternion matrix equation into the real domain.Constrained equations are established,and the conjugate gradient iterative algorithm is established to obtain the quaternion cyclic solution of the equation.Numerical examples verify the convergence and feasibility of the given algorithm.3.The Hankel solution of the quaternion matrix equation(AXB,CXD)=(E,F)is studied.Using the complex decomposition of the quaternion matrix,the quaternion Hankel matrix is vectorized and characterization,and the unconstrained equivalence equations are obtained.The conjugate gradient iterative algorithm is established to obtain the Hankel matrix solution of the equation.The convergence and feasibility of the algorithm are given by numerical examples.