Quaternionic Matrices: Inversion and Determinant ∗

We discuss the Schur complement formula for quaternionic matrices, M , and give an efficient method to calculate the matrix inverse. We also introduce the functional D[M ] which extends to quaternionic matrices the non-negative number |det[M ]|. 1. Introduction. Much of the spectral theory of complex matrices does not extend to quaternion matrices without further modifications [1, 2, 3]. In particular, the determinant cannot be defined, and the eigenvalues [4] have several possible extensions which do not necessarily respect the fundamental theorem of algebra [5, 6, 7]. Quaternionic mathematical structures have recently appeared in the physical litterature mainly in the context of quantum mechanics [8, 9, 10] and group theory [11, 12]. The relevance of quatenionic calculus to several iussues in theoretical phycis has rekindled the interest in quaternionic linear algebra. A usefull thecnical procedure in approaching quaternionic applications in physics has been the replacement of quaternionic matrices by complex matrices of double size [13, 14], and the consequent spectral analysis of the latter matrix [4, 15, 16]. This indirect procedure, usually, suffices to solve the problem at hand, but it sheds little light on the structure of quaternionic matrices. We aim to discuss inversion and determinant for quaternionic matrices in view of new classifications of quaternionic groups and possible applications in grand unification theory [17, 18]. The main results of this paper can be summarized by Inversion. The formula M −1 = Adj[M ]/det[M ] cannot be generalized for quater-nionic matrices. Nevertheless, the Schur complement formula provides an efficient method to calculate the matrix inverse. Determinant. The determinat for n × n quaternionic matrices, M , cannot be defined in a consistent way. Instead, the non-negative number |det[M ]| can be extended to quaternionic matrices by the functional D[M ] defined in terms of quaternionic M-entries.