The quaternionic determinant

The determinant for complex matrices cannot be extended to quaternionic matrices. Instead, the Study determinant and the closely related q-determinant are widely used. We show that the Study determinant can be characterized as the unique functional which extends the absolute value of the complex determinant and discuss its spectral and linear algebraic aspects. 1. Introduction. Quaternionic linear algebra is attracting growing interest in theoretical physics [1]-[5], mainly in the context of quantum mechanics and field theory [6]. Quaternionic mathematical structures have recently appeared in discussing eigenvalue equations [7, 8], group theory [9, 10] and grand unification model [11, 12] within a quaternionic formulation of quantum physics. The question of extending the determinant from complex to quaternionic matrices has been considered in the physical literature [4]-[6]. The possibility of such an extension has been contemplated by Cayley [13], without much success, as early as 1845. A canonical determinant functional was introduced by Study [14] and its properties axiomatized by Dieudonné [15]. The details can be found in the excellent survey paper of Aslasken [16]. Study's determinant is denoted as Sdet, and up to a trivial power factor, is identical to the q-determinant, det q , found in most of the literature [17] and to Dieudonné's determinant, denoted as Ddet. Study's determinant is closely related to the q-determinant and to Dieudonne's determinant. Specifically, det q = Sdet 2 = Ddet 4. In these works, Sdet was considered as a generalization of the determinant, det, in the sense that the two functionals share a common set of axioms. Specifically, Sdet is the unique, up to a trivial power factor, functional F : H n × n which satisfies the following three axioms: 1. F (A) = 0 if and only if A is singular; 2. multiplicativity: F (AB) = F (A)F (B); 3. F (I + rE ij) = 1 for i = j and r ∈ H; see [16]. However, Sdet does not truly extend det. Indeed, the two functionals do not coincide on complex matrices, since the former is nonnegative while the latter is truly complex. In this paper we show that Sdet does extend the nonnegative functional |det|, namely the two functionals coincide for complex matrices. More precisely, we show the following: