Yes, that is the purpose of a signature.
Where or When or Why do I sign?
I sign to say that I was present at the time of signing, I sign a document to say that I agree to something written above it, I sign a bank cheque to authenticate the transaction.
You can see that my signature is useful to represent me in my absence. So it
is my representation.
Signature is a special type of a handwriting. When we write the handwriting comes naturally. But when we sign we are reproducing a chosen formation which has been practiced several times. When we decide to make a graphic or a word as our signature we are aware that its going to represent us in our absence. Thus we choose a signature unknowingly which is like our personality.
Signature is your public image.