四元数矩阵方程AX=B的酉矩阵最小二乘解

The Unitary Least-Square Solution of the Quaternion Matrix Equation AX=B

以UQn×n 表示四元数酉矩阵的全体。本文给出了四元数矩阵方程AX=B在UQn×n 中最小二乘解的表达式 ,以及AX=B在UQn×n 中有解的充分必要条件 ,通解的表达式。

Let UQ^(n×n) be the set of n×n unitary qaternion matrices.This paper discusses the following problems: Problem 1 Given A,B∈Q^(m×n),fand (X)∧∈UQ^(n×n),such that ‖A(X)∧-B‖_F=minX∈UQ^(n×n)‖AX-B‖_F; Problem 2 Given A,B∈Q^(m×n), find X∈UQ^(n×n),such that AX=B. The expression of the solution to Problem 1 is provided.The necessary and sufficient condition of Problem 2 are presented.The general solutions of Problem 2 is given.