FAQ: How do you show that exp(T S inv(T)) = T exp(S) inv(T)?

Posted by Murray 08:11, 22 April 2006 (PDT)
This can be shown by substitution into the definition of the matrix exponential:

$\left.exp(T S T^{-1}) \right.$	$= I + (T S T^{-1}) + \frac{1}{2} (T S T^{-1})^2 + \frac{1}{3!} (T S T^{-1})^3 + \dots$
	$= I + (T S T^{-1}) + \frac{1}{2} \left( T S T^{-1} T S T^{-1} \right) + \frac{1}{3!} \left( T S T^{-1} T S T^{-1} T S T^{-1} \right) + \dots$
	$= I + (T S T^{-1}) + \frac{1}{2} \left(T S^2 T^{-1}\right) + \frac{1}{3!} \left(T S^3 T^{-1}\right) + \dots$
	$= T \left( S + \frac{1}{2} S^2 + \frac{1}{3!} S^3 + \dots \right) T^{-1}$
	$= \left. T exp(S) T^{-1} \right.$