The Matrix Logarithm and the Continuization of a Discrete Process

The inverse problem of the discretization for a linear system is solved. It ties in with the basic theory of the matrix logarithm (which is multivalued). Conditions, in terms of the elementary divisors, are presented under which a given matrix has a real logarithm. It is further shown that an n-th order discrete system can always be "continuized" by a minimal real system of order between n and 2n. Applications to multi-rate control and interpolation are give.