ABSTRACT

In this paper, we consider the quaternion matrix equation $X-A\hat{X}B=C$, and study its minimal norm least squares solution, j-self-conjugate least-squares solution and anti-j-self-conjugate least-squares solution. By the real representation matrices of quaternion matrices, their particular structure and the properties of Frobenius norm, we convert above least-squares problems into corresponding problems of real matrix equations. The final results of the expressions only involve real matrices, and thus, the corresponding algorithms only involve real operations. Compared with the existing results, they are more convenient and efficient, which are also illustrated by the last two numerical examples.

摘要

在本文中，我们考虑四元数矩阵方程$X-\mathrm{一个}\hat{X}乙=C$，并研究其最小范数最小二乘解、j-自共轭最小二乘解和反-j-自共轭最小二乘解。通过四元数矩阵的实数表示矩阵、它们的特殊结构以及Frobenius范数的性质，我们将上述最小二乘问题转化为实数矩阵方程的对应问题。表达式的最终结果只涉及实数矩阵，因此相应的算法只涉及实数运算。与现有结果相比，它们更加方便和高效，最后两个数值例子也说明了这一点。