Amalek was the grandson of Edom, the son of Isaac: “Now Timna was the concubine of Eliphaz, Esau's son, and she bore Amalek to Eliphaz.” (Genesis 36:12 NKJV)

 The word translated as “Amalek” means “dweller in a valley.” (Amaleq OT:6002 Brown Driver & Briggs Hebrew Lexicon) Thus the statement that the Amalekites dwelt in a valley restates the obvious: “Now the Amalekites and the Canaanites dwell in the valley.” (Numbers 14:25 NKJV) “These were the chiefs of the sons of Esau. The sons of Eliphaz, the firstborn son of Esau, were Chief Teman, Chief Omar, Chief Zepho, Chief Kenaz, Chief Korah, Chief Gatam, and Chief Amalek. These were the chiefs of Eliphaz in the land of Edom. They were the sons of Adah.” (Genesis 36:15-16 NKJV)
 There is a reference to the Amalekites BEFORE the birth of the person named Amalek, the descendant of Esau. “In the fourteenth year Chedorlaomer and the kings that were with him came … and came to En Mishpat (that is, Kadesh), and attacked all the country of the Amalekites.” (Genesis 14:5-7 NKJV) However, the reference is to “the COUNTRY of the Amalekites” rather than the actual people called the Amalekites. The reference to the area of the Amalekites at the time of Abraham may merely be a way of describing an ancient area in terms that would be understandable to the people living at the time when Genesis was written, namely at the time of Moses.
The dwelling place of the descendants of Amalek, the grandson of Esau, is south and east of the territory of Judah.
 In the land of the south - “The Amalekites dwell in the land of the South; the Hittites, the Jebusites, and the Amorites dwell in the mountains; and the Canaanites dwell by the sea and along the banks of the Jordan.” (Numbers 13:29 NKJV)
 In Shur, which is east of Egypt - “And Saul attacked the Amalekites, from Havilah all the way to Shur, which is east of Egypt.” (1 Samuel 15:7 NKJV) “And David and his men went up and raided the Geshurites, the Girzites, and the Amalekites. For those nations were the inhabitants of the land from of old, as you go to Shur, even as far as the land of Egypt.” (1 Samuel 27:8 NKJV)
 Near Kadesh – “Then they turned back and came to En Mishpat (that is, Kadesh), and attacked all the country of the Amalekites.” (Genesis 14:7 NKJV)
 Near the Idumean mountains (near Edom) – The Jamieson, Fausset, and Brown Commentary states, “On the other side of the Idumean mountain at whose base they were then encamped.” (Numbers 14:25 Jamieson, Fausset, and Brown Commentary)
Any reference to the Amalekites living in the central or northern part of Israel AFTER the time of Amalek, the grandson of Esau, may indicate only that the group became so large that they expanded into new areas.

The Amalekites interacted with the Jews for many centuries
 At the time of Moses when the Israelites first left Egypt – “Now Amalek came and fought with Israel in Rephidim. And Moses said to Joshua, ‘Choose us some men and go out, fight with Amalek. Tomorrow I will stand on the top of the hill with the rod of God in my hand.’ So Joshua did as Moses said to him, and fought with Amalek. And Moses, Aaron, and Hur went up to the top of the hill. And so it was, when Moses held up his hand, that Israel prevailed; and when he let down his hand, Amalek prevailed.  But Moses' hands became heavy; so they took a stone and put it under him, and he sat on it. And Aaron and Hur supported his hands, one on one side, and the other on the other side; and his hands were steady until the going down of the sun. So Joshua defeated Amalek and his people with the edge of the sword. Then the Lord said to Moses, ‘Write this for a memorial in the book and recount it in the hearing of Joshua, that I will utterly blot out the remembrance of Amalek from under heaven.’ And Moses built an altar and called its name, The-Lord-Is-My-Banner; for he said, ‘Because the Lord has sworn: the Lord will have war with Amalek from generation to generation.’” (Exodus 17:8-16 NKJV)
 At the time of Moses when the Israelites first attempted to enter Canaan – “Then Moses told these words to all the children of Israel, and the people mourned greatly. And they rose early in the morning and went up to the top of the mountain, saying, ‘Here we are, and we will go up to the place which the Lord has promised, for we have sinned!’ And Moses said, ‘Now why do you transgress the command of the Lord? For this will not succeed. Do not go up, lest you be defeated by your enemies, for the Lord is not among you. For the Amalekites and the Canaanites are there before you, and you shall fall by the sword; because you have turned away from the Lord, the Lord will not be with you.’ But they presumed to go up to the mountaintop. Nevertheless, neither the ark of the covenant of the Lord nor Moses departed from the camp. Then the Amalekites and the Canaanites who dwelt in that mountain came down and attacked them, and drove them back as far as Hormah.” (Numbers 14:39-45 NKJV)
 At the time of the Ehud the Judge of Israel: “And the children of Israel again did evil in the sight of the Lord. So the Lord strengthened Eglon king of Moab against Israel, because they had done evil in the sight of the Lord. Then he gathered to himself the people of Ammon and Amalek, went and defeated Israel, and took possession of the City of Palms. So the children of Israel served Eglon king of Moab eighteen years. But when the children of Israel cried out to the Lord, the Lord raised up a deliverer for them: Ehud the son of Gera, the Benjamite, a left-handed man.” (Judges 3:12-15 NKJV)
 At the time of Gideon, the judge of Israel – “Then the Lord said to Gideon, ‘By the three hundred men who lapped I will save you, and deliver the Midianites into your hand. Let all the other people go, every man to his place’ … Now the Midianites and Amalekites, all the people of the East, were lying in the valley as numerous as locusts; and their camels were without number, as the sand by the seashore in multitude.” (Judges 7:7-12 NKJV)
 At the time of Saul: “Thus says the Lord of hosts: ‘I will punish Amalek for what he did to Israel, how he ambushed him on the way when he came up from Egypt. Now go and attack Amalek, and utterly destroy all that they have, and do not spare them. But kill both man and woman, infant and nursing child, ox and sheep, camel and donkey.’ So Saul gathered the people together and numbered them in Telaim, two hundred thousand foot soldiers and ten thousand men of Judah. And Saul came to a city of Amalek, and lay in wait in the valley … And Saul attacked the Amalekites, from Havilah all the way to Shur, which is east of Egypt.”  (1 Samuel 15:2-7 NKJV)