[Solved] What is the mathematical difference between a transfer function and a state transition matrix?

LandonZeKepitelOfGreytBritn Asks: What is the mathematical difference between a transfer function and a state transition matrix?
I do know what the difference in meaning is between a transfer function (output over the input) and a state transition matrix $Phi$ (describes the unforced response of the system). Yet when looking closer at the mathematics to me it seems like they are both the same. Could someone clarify?

$$dot{q} = Aq(t)+Bu(t)$$ where A is the state matrix, q the state vector, B the input matrix and u the input vector.

$$sQ(s) = AQ(s) + BU(s)$$ $$ sQ(s)- AQ(s) = BU(s)$$ $$ Q(s)(sI-A) = BU(s) $$ $$ Q(s) = (sI-A)^{-1}BU(s) $$

where $(sI-A)^{-1} = Phi$ ie the state transition matrix.

$$ Q(s) = Phi BU(s)$$ $$ Phi = frac{Q(s)}{BU(s)}$$

$frac{Q(s)}{BU(s)}$ looks to me like the representation of a transfer function and based on the math it looks like the state transition matrix in fact equals a transfer function, yet that doesn’t correspond to the interpretation I have of both of those things. Could somebody please elaborate a bit on that?

Ten-tools.com may not be responsible for the answers or solutions given to any question asked by the users. All Answers or responses are user generated answers and we do not have proof of its validity or correctness. Please vote for the answer that helped you in order to help others find out which is the most helpful answer. Questions labeled as solved may be solved or may not be solved depending on the type of question and the date posted for some posts may be scheduled to be deleted periodically. Do not hesitate to share your response here to help other visitors like you. Thank you, Ten-tools.