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:

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

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