Finite-dimensional Preliminaries

In this chapter we discuss the finite-dimensional aspects of quaternionic analysis which are needed in the sequel. In the first section we survey the main properties of quaternions. Then we consider quaternionic polynomials and we discuss their zeros. In the third section we discuss quaternionic matrices and basic definitions such as adjoint, transpose, and inverse.We introduce the map χ which to any quaternionic square matrix associates a complex matrix of double size. In particular, this map is also defined for quaternions. We discuss the eigenvalue problem and show that it is associated with the notion of left and right spectrum and with the so-called Sspectrum. Finally, we present the Jordan decomposition of a matrix. In the fourth and last section we consider some matrix equations which appear in the sequel.