Jamieson, Fausset and Brown on 1 Samuel 15:2-3

2-3. Amalek--the powerful tribe which inhabited the country immediately to the eastward of the northern Cushites. Their territory extended over the whole of the eastern portion of the desert of Sinai to Rephidim--the earliest opponent (De 25:18; Ex 17:8-16)--the hereditary and restless enemy of Israel (Nu 14:45; Jud 3:13; 6:3), and who had not repented (1Sa 14:48) of their bitter and sleepless hatred during the five hundred years that had elapsed since their doom was pronounced. Being a people of nomadic habits, they were as plundering and dangerous as the Bedouin Arabs, particularly to the southern tribes. The national interest required, and God, as King of Israel, decreed that this public enemy should be removed. Their destruction was to be without reservation or exception.

I remember--I am reminded of what Amalek did--perhaps by the still remaining trophy or memorial erected by Moses (Ex 17:15, 16).