Attempts to define a matrix exponential over (as much as possible) general fields

Question

Given a $n \times n$ matrix $A$ over the complex numbers, the exponential of $A$ is defined as $$\exp(A) := \sum_{k = 1}^\infty \frac1{k!} A^k , \qquad \tag{$\star$}\label{468645_star}$$ where convergence is meant with respect to an arbitrary norm on the space of $n \times n$ complex matrices (since such norms are all equivalent). Then $\exp(\cdot)$ has many nice properties such as satisfying the identity $\exp(A + B) = \exp(A) \exp(B)$ whenever $A$ and $B$ are commuting.

I wonder if somebody tried to define an exponential function for matrices over an arbitrary field $\mathbb{K}$. Of course there are several obstructions in doing so: one cannot speak of the convergence of \eqref{468645_star} if $\mathbb{K}$ is missing a topology; if $\mathbb{K}$ has positive characteristic then almost all the fractions $1/k!$ of \eqref{468645_star} are not defined…. However, there can be also some workarounds, such as assuming $A$ nilpotent (and with small order) so that the series in \eqref{468645_star} is a finite sum (with the fractions $1/k!$ all defined)… and maybe Jordan–Chevalley decomposition makes possible to extend the definition to all matrices (or one accepts an exponential defined only for some matrices).

Did somebody try to define a matrix exponential for more general fields than the complex numbers and how far did she get? I'm particularly interested in the case of positive characteristic (finite fields would already be interesting).

Thank for any reference/suggestion.

Answer 1

You can give an equivalent definition of the matrix exponential. or any matrix function $f$, using the Jordan form. If $A = VJV^{-1}$, and $J$ is the direct sum of diagonal blocks of the form $$ J_i = \begin{bmatrix} \lambda_i & 1\\ & \lambda_i & \ddots\\ & & \ddots & 1\\ & & & \lambda_i \end{bmatrix}, $$ then $f(A) = Vf(J)V^{-1}$, where $f(J)$ is the direct sum of Toeplitz triangular blocks of the form $$ J_i = \begin{bmatrix} f(\lambda_i) & f'(\lambda_i) & \frac{f''(\lambda_i)}{2} & \dots & \frac{f^{(k)}(\lambda_i)}{k!}\\ & \ddots & \ddots & & \vdots\\ & & \ddots &\\ & & & f(\lambda_i)& f'(\lambda_i)\\ & & & & f(\lambda_i) \end{bmatrix}. $$ For the exponential, the derivatives are trivial, so any field for which you can define both the scalar $\exp$ and the Jordan form works. Algebraically closed is the only condition needed for the Jordan form, if I am not missing anything. [EDIT: Also the fractions $\frac{1}{k!}$ are a problem in non-zero characteristic, unfortunately. Note, though, that they are needed only in cases where $A$ has a Jordan block of size larger than $\operatorname{char}(\mathbb{F})$, so at least for some matrices you can still apply the definition.]

Also, at this point, I imagine that you can extend the definition also to a non-algebraically-closed field $\mathbb{F}$: work in its algebraic closure $\overline{\mathbb{F}}$, and note that $f(A)$ must have entries in $\mathbb{F}$ since it does not depend on the choice of the closure.

Alternatively, a matrix function $f(A)$ can also be defined using Hermite interpolation: take a polynomial $p(x)$ such that $p^{(j)}(\lambda_i) = f^{(j)}(\lambda_i)$, for each eigenvalue $i$ and each $j=0,1,\dots, m_a(\lambda_i)$ (algebraic multiplicity of $\lambda_i$); then $f(A) = p(A)$. This definition works out of the box for the exponential.

Note that all these definitions coincide with the limit of the series in cases where it is defined.

For more information on these techniques, you can check Chapter 1 in Higham's excellent book Functions of matrices.

EDIT: note that the second method has the same requirements on denominators as the first one, since the formulas to define Hermite interpolants use $\frac{f^{(k)}(\lambda_i)}{k!}$ internally; see e.g. the divided differences method on Wikipedia.