Alternatives to the Matrix Exponential

Question

The exponential of the matrix $At$ is related to linear systems of ordinary differential equations. It is given by the concise yet apparently somewhat arbitrary formula $e^{At} = Y(t)Y(0)^{-1}$, where the order of operations between inverting the second $Y$ and evaluating it at $0$ doesn't matter.

Oftentimes in linear algebra, there are multiple ways to view basic operations (column and row approaches to multiplication), or multiple ways operations could be defined, with one arbitrary choice being declared correct. Do expressions similar to the actual $e^{At}$ formula such as $$Y(0)Y(t)^{-1}$$$$Y(t)^{-1}Y(0)$$$$Y(0)^{-1}Y(t)$$ relate to the ODE system as well, or is there something special about the actual formula that conveys informations about the system?

Answer 1

I do not think you define the exponential of a matrix via systems of equations (although that is possible of course). You define it independently of any ODE as a series $$ e^A=I+A+\frac{A^2}{2!}+\dots $$ which is convergent for any matrix $A$. Then you prove some basic properties of this function and, in particular, you prove that $$ \frac{d}{dt}e^{At}=Ae^{At} $$ a property that is telling you that $e^{At}x_0$ is a solution of the differential system $x'=Ax$, and all you need to adjust is the initial value, so you need $x_0=x(0)$. In other words, $$ x(t)=e^{At}x(0), $$ or, for any system of solutions $Y(t)$ (in particular for the fundamental system), $$ Y(t)=e^{At}Y(0) $$ from which your relation follows. But I do not think the distinction between representations as $Y(t)Y^{-1}(0)$ and similar ones is relevant.

Answer 2

First,there is no ncessary connection between systems of differential equations and the matrix exponential. Whenever I taught a course in linear algebra that had solving systems of differential equations as an application of diagonalization by similarity, I avoided getting into matrix exponentials, which seemed, to my purely subjective view, to be an unnecessarry red herring. Second, the power series definition for the exponential function of anything works perfectly well over an arbitrary real or complex Banach algebra, in particular for a square real or complex matrix. Then the formulae for the exponential of a sum works fine for two commuting matrices and the differential equations formula is an easy consequence of the definitions. So, my opinion is :Keep the ideas separate at first and bring them together as a delightful bonus!