Why did the ancient Jews believe the messiah would be a great warrior who would establish the kingdom through military conquest?

The ancient Jews believed the messiah would be a great warrior who would establish the kingdom through military conquest because the people's expectation of a savior became bound up with the notion of a new nation, political kingship.

What were Matthew's intentions?

Matthew's intentions were to show all the Jewish readers that Jesus was the Messiah that they had all been waiting for. "The son of David, the son of Abraham" (1:1).