Quaternionic Determinants

The classical matrix groups are of fundamental importance in many parts of geometry and algebra. Some of them, like Sp.n/, are most conceptually defined as groups of quaternionic matrices. But, the quaternions not being commutative, we must reconsider some aspects of linear algebra. In particular, it is not clear how to define the determinant of a quaternionic matrix. Over the years, many people have given different definitions. In this article I will try to discuss some of these. I would like to thank Jon Berrick, P. M. Cohn, Soo Teck Lee and the referee for help with improving this paper. Let us first briefly recall some basic facts about quaternions. The quaternions were discovered on October 16 1843 by Sir William Rowan Hamilton. (For more on the history, I recommend [19, 31, 47, 48].) They form a noncommutative, associative algebra over R: