e^{tA} = \sum\limits_{n = 0}^\infty {\frac{{\left( {tA} \right)^n }}
{{n!}}} = \sum\limits_{n = 0}^\infty {\frac{{t^n \left( {TJT^{ - 1} } \right)^n }}
{{n!}}} = \sum\limits_{n = 0}^\infty {\frac{{t^n TJ^n T^{ - 1} }}
{{n!}}} = T\left( {\sum\limits_{n = 0}^\infty {\frac{{t^n J^n }}
{{n!}}} } \right)T^{ - 1} = Te^{tJ} T^{ - 1}