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• The "northern kingdom" refers to Israel, led by the tribe of Ephraim, after the House of Israel was split into two houses.
• The king of the north refers to Kings of Babylon, specifically Tyrus Nebuchadrezzar in Ezekiel 26:7, and to Nebuchadrezzar (Nebuchadrezzar) in Jeremiah 25:9.

Notes about the Northern Nations:

• In Jeremiah 50, though, another northern nation is mentioned "For out of the north there cometh up a nation against her (Babylon), which shall make her land desolate..." (Jer 50:3)
• Also: "For, lo, I will raise and cause to come up against Babylon an assembly of great nations from the north country" (Jer 50:9)
• One of the northern nations must refer to Hamadan or one of the places where King Cyrus is said to have ruled from in Persia - modern day Iran.