Plane waves in linear homogeneous media. I

A wide class of linear one dimensional propagation phenomena in a homogeneous medium can be described by a vector equation containing a time derivative and a spatial operator, which after a Fourier transformation amounts into a multiplication by a holomorphic matrix. This class of wave equations admits a general treatment concerning a representation of the solution of the initial value problem, mode-decomposition, quadratic conservation laws, group velocity and stability. This first paper provides the mathematical preliminaries for such a treatment. A theory of holomorphic matrices is developed which includes a.o. a discussion of the analytic behaviour of eigenvalues and eigenvectors and a commutator theory. The "differentiation-like operator" concept is introduced.