Definition for a topological field
Suppose is a topological field. The matrix exponential, denoted , is defined as the map from (a suitable subset of) the set of all matrices over , denoted , to the set of invertible matrices over , i.e., the general linear group . It is defined as:
More formally, it is the limit of the partial sums:
where the limit is taken entry-wise on the matrices with respect to the field topology. Note the following facts:
- For the field of real numbers as well as the field of complex numbers (equipped with the usual topologies), the matrix exponential is defined for all matrices.
- For the field of p-adic numbers, the matrix exponential is defined for all matrices in which all entries are -multiples of elements of the -adic integers (for odd). For , we need all entries to be 4 times elements in the p-adic integers.
Definition for nilpotent matrices
Suppose is any field and is a nilpotent matrix in with for some . Suppose further that the characteristic of is either equal to zero or at least equal to . Then, we define: