When a question 'Where Swami Exists' or 'What meaning the expression Swami conveys'  is thrown open, we may have to ponder for a while for an answer.
Swami
 exists in Vaikunta. He lives in our 'hirudaya or heart'. He also lives 
in the Temple. That is why, we travel long distances to visit famous and
 sacred shrines as we believe they are the birth places of 
 Swami or connected with some notable 'sthalapuranams'. We get a relief 
that we have fulfilled the commitments made long ago by worshiping Swami
 in those places. 
As Swami exists in the heart, may be the 
reason
 for calling him 'hirudayakamalavasa', we think of him by feeling his 
existence there and then do 'avahanam' in an idol or Murthy by means of a simple 
'sankalpa'. 
It is also believed that Swami exists in Kailash also as 
in Vaikunta - another name for Vaikunta is 'Paramapadam'. To quote Veda "
 Tat Vishno ;  Paramam  Padam ". Vaishnavites refer when a person dies 
that he ascends to Vaikunta.
A title is given to Swami as 'Purushothma' 
with reference to his place of living in Vaikunta. The meaning of 
Purushothama becomes clear when we split it as Purusha + Uthama. In Tamil
 Purushothaman gets the meaning as "PERUM + Al", that is "PERUMAL". 
'Uth' in Uthama means in Tamil as 'Periyavar'. Uthamar is used as a 
qualifying term to refer a person as respectable or adorable. By fixing 
the prefix Purusha before Uthama, the resulting word is 'Purushothama' 
which symbolically refer to Narayana who is undoubtedly the 
'Uthama-Purusha'. 
The word swami has 'swam' in it. 'Swam' in
 Sanskrit means 'possession' or 'property'. Swami signifies the one who 
possesses. What or which property Swami owns? Jagat Guru Kanchi Kamakoti
 Sri Chandrasekarendra Saraswati Sankaracharya Swamigal had explained 
that whenever we refer to our family deity, it means the deity who owns 
us or we who belong to him.
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