Matrix exponential

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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)!}