# Matrix exponential

## Definition

### 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:

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