将Frobenius求逆推广到四元数矩阵
Generalizing Frobenius Inversion to Quaternion Matrices
AI总结：
本研究提出一种对应复矩阵Frobenius算法的四元数矩阵求逆算法，理论上证明其复求逆次数最少、效率优于现有方法，实验中随机矩阵表现更优，可提升Strassen型递归算法的实用性。
摘要：
本文推导并分析了一种四元数矩阵求逆算法，该算法是复矩阵求逆Frobenius算法的对应版本。理论层面，我们证明该算法比现有其他方法效率更高，且在复矩阵求逆次数最少的意义下是最优的。实践层面，在随机生成的矩阵上，该算法表现优于现有算法。我们认为，该算法可为四元数矩阵递归分解过程提供最快的基例，从而提升递归Strassen型算法的实用价值。
英文摘要：
In this paper we derive and analyze an algorithm for inverting quaternion matrices. The algorithm is an analogue of the Frobenius algorithm for the complex matrix inversion. On the theory side, we prove that our algorithm is more efficient that other existing methods. Moreover, our algorithm is optimal in the sense of the least number of complex inversions. On the practice side, our algorithm outperforms existing algorithms on randomly generated matrices. We argue that this algorithm can be used to improve the practical utility of recursive Strassen-type algorithms by providing the fastest possible base case for the recursive decomposition process when applied to quaternion matrices.
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