1

It is known that, in general, for two matrices $A$ and $B$, $\mathrm{e}^A \mathrm{e}^B \ne \mathrm{e}^{A+B}$. Is there a word to denote this? I'm looking for something like "nonlinear", but that would mean that $\mathrm{e}^{A+B} \ne \mathrm{e}^A+\mathrm{e}^B$.

Martin Brandenburg
  • 181,922
Jellby
  • 299

1 Answers1

4

The term you're looking for is homomorphism (specifically, monoid homomorphism).

Fix $n>1$, and let $M_{add}$ and $M_{mult}$ be the monoids of $n$-by-$n$ (real) matrices with respect to matrix addition and matrix multiplication respectively. The statement that $e^Ae^B\not=e^{A+B}$ in general is the observation that $X\mapsto e^X$ is not a homomorphism from $M_{add}$ to $M_{mult}$. In general, a homomorphism between two monoids $(A,\star)\rightarrow (B,*)$ is a function $f:A\rightarrow B$ such that $f(a \star a')=f(a)*f(a')$ and which additionally sends the identity element of $A$ to the identity element of $B$ (an earlier version of this question made a really stupid error at this point - that's what happens when you try to do math at 4:30 AM!).

Noah Schweber
  • 260,658
  • 2
    You took the wrong neutral element. We have $e^0 = 1$. The monoid structure is relevant here, or actually the group structure. By general theory $e^a$ is invertible, so we have a map $\exp : \mathrm{M}n \to \mathrm{GL}_n$. This is the "natural codomain", also when thinking about the [generalization to Lie groups](https://en.wikipedia.org/wiki/Exponential_map%28Lie_theory%29?wprov=sfla1). – Martin Brandenburg Nov 30 '24 at 14:18
  • @MartinBrandenburg Oh my goodness. Fixed! (Note to self: don't do math at 4:30 AM ... at least, not where others can see the results!) – Noah Schweber Nov 30 '24 at 18:37
  • No worries, you are not alone. – Martin Brandenburg Nov 30 '24 at 19:01