Proving Matrix exponential property

[kidsasd987]

this is not a homework question, I just want to make sense of the equation here.
Assuming matrix A is diagonal,

If A_hat=T'AT where T' is an inverse matrix of T.

e^(A_hat*t)=T'e^(At)T
which implies,
e^(T'AT*t)=T'e^(At)T

we know that e^(At) is a linear mapping, therefore if we convert f to some linear transformation P,
PT'AT=T'PAT (not sure if this step is correct) this condition should be always true, but why?can anyone provide me a short proof of this?

 

[DuckAmuck]

The Taylor expansion of e^x is:
$$e^x = 1 + x + x^2/2! + x^3/3! + x^4/4! + ...$$
So when x is a matrix T'AT, you get a lot of cancellations
For example in the x^4/4! term, you get
$$(T'AT)^4/4! = T'AT T'AT T'AT T'AT /4! = T' A^4 T /4!$$
This same kind of TT' = 1 cancellation happens in every term until you are left with T' on the left and T on the right of every term.
So you get:
$$T' (1 + A + A^2/2! + A^3/3! + A^4/4! + ...) T = T'e^A T$$