Matrix exponential

Definition for a topological field
Suppose $$K$$ is a topological field. The matrix exponential, denoted $$\exp$$, is defined as the map from (a suitable subset of) the set of all $$n \times n$$ matrices over $$K$$, denoted $$M(n,K)$$, to the set of invertible $$n \times n$$ matrices over $$K$$, i.e., the general linear group $$GL(n,K)$$. It is defined as:

$$\exp(X) := \sum_{k=0}^\infty \frac{X^k}{k!} = I + X + \frac{X^2}{2!} + \frac{X^3}{3!} + \dots$$

More formally, it is the limit of the partial sums:

$$\exp(X) := \lim_{m \to \infty} \sum_{k=0}^m \frac{X^k}{k!}$$

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 $$p$$-multiples of elements of the $$p$$-adic integers (for $$p$$ odd). For $$p = 2$$, we need all entries to be 4 times elements in the p-adic integers.

Definition for nilpotent matrices
Suppose $$K$$ is any field and $$X$$ is a $$n \times n$$ nilpotent matrix in $$K$$ with $$X^m = 0$$ for some $$m$$. Suppose further that the characteristic of $$K$$ is either equal to zero or at least equal to $$m$$. Then, we define:

$$\exp X = \sum_{k=0}^{m-1} \frac{X^k}{k!} = I + X + \frac{X^2}{2!} + \dots + \frac{X^{m-1}}{(m-1)!}$$

Facts

 * Matrix exponential commutes with conjugation
 * Exponential of sum of commuting matrices is product of exponentials
 * Eigenvalues of matrix exponential are exponentials of eigenvalues
 * Determinant of matrix exponential is exponential of trace