# [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?

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