State space and matrix exponential

Hi all, i spent some time with this question How does the stability of a state space system affect the corresponding matrix exponential? How does its controllability and/or
observability?
I deduced the following answer The eigen values of the state transition matrix is given by the exponential of the state matrix, and the stability of the depends on the eigen values of the state transition matrix. If the system is stable, then the matrix exponential will be finite. Controllability and/or observability won't affect the matrix exponential, because the matrix exponential is unique. Please correct me if i am wrong. jc
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