This isn't something that's unique in Indian, or indeed, Tamil history by any means; the Pallavas faced it too, after the death of Parameswara Varman in (probably) 731 CE. Nandivarman II (called Pallavamalla), who was a matrilineal descendant was invited back from Kambhudesa (Cambodia/Vietnam) to become the king (http://en.wikipedia.org/wiki/Nandivarman_II)
Also, wasn't Vikrama Cholan the son of Kulothunga I, and not the son of an(other) Eastern Chalukya?
Very good analysis, I have a small doubt what would be the langugage of the eastern chalukyas was it telugu or kannada , did kulothunga 1 spoke telugu or tamil
Chalukyas were Telugu, the Epigraphic Evidences given below from No: 2 to 6 are in Telugu. Kulothunga - 1's queen, mother, and grandmother, were all Chola Princesses speaking Tamil. As his Paternal side were Chalukyas, Kulothunga Chola - 1 would have definitely been versant in Telugu too. He ruled Chola country for 52 years from Gangaikonda Cholapuram in Tamil Nadu. Hence it is quite possible that he would have used Telugu & Tamil as his spoken language prior to becoming the Chola king, and essentialy in Tamil and sparingly in Telugu after he became the king of the Chola country where all people spoke Tamil.