Quaternion equality constrained least squares (QLSE) problems have attracted extensive attention in the field of mathematical physics due to its applicability as an extremely effective tool. However, the knowledge gap among numerous QLSE problems has not been settled now. In this paper, by using quaternion SVD (Q-SVD) and the equivalence of the QLSE problem and Karush-Kuhb-Tucker (KKT) equation, we obtain some equations about the matrices in the general solution of the QLSE problem. Using these equations, an equivalent form of the solution of the QLSE problem is obtained. Then, applying the special structure of real representation of quaternion, we propose a real structure-preserving algorithm based on Q-SVD. At last, we give numerical example, which illustrates the effectiveness of our algorithm.

四元数等式约束最小二乘（QLSE）问题由于其作为一种极其有效的工具而在数学物理学领域引起了广泛关注。但是，许多QLSE问题之间的知识鸿沟目前尚未解决。本文通过使用四元数SVD（Q-SVD）以及QLSE问题的等价性和Karush-Kuhb-Tucker（KKT）方程，在QLSE问题的一般解中获得了一些关于矩阵的方程。使用这些方程式，可以获得QLSE问题的等价形式。然后，运用四元数实数表示的特殊结构，提出了一种基于Q-SVD的实数保留算法。最后，给出数值例子，说明了算法的有效性。