Properties of Exponential Matrix

Question

One of the properties is that $e^{{\bf A}+{\bf B}}\neq e^{\bf A}e^{\bf B}$ unless ${\bf AB}$$={\bf BA}$. Can someone please explain how exactly commutativity matters in this case? I'm guessing it has something to do with series multiplication?

Answer 1

I guess you'll want to see the Trotter product formula. For any complex $A,B$ matrices we have $$ \exp ( A + B ) = \lim_{N\to \infty} \left [ \exp \left ( \frac{A}{N} \right) \exp \left ( \frac{B}{N} \right ) \right ] ^N $$ You'll have the equality of $e^{A+B} = e^{A} e^{B}$ when you expand the product and you have commutativity.

Answer 2

The exponential of a matrix is defined by the Taylor Series expansion

The basic reason is that in the expression on the right the $A$s appear before the $B$s but on the left hand side they can be mixed up . Expanding to second order in $A$ and $B$ the equality reads

$$ e^{A+B} =e^A e^B $$ $$\implies 1+A+B+\frac 12 (A^2+AB+BA+B^2)=(1+A+\frac 12 A^2)(1+B+\frac 12B^2)+\text{ higher order terms }$$

The constants and the first order terms cancel. Truncating at second order we get $$\frac 12 (AB+BA)=AB \implies AB=BA$$