Bhagavad Gita, the great digest of the upanishads, has to say about this:

 “Bodies are said to die, but That which possesses the body is eternal. It cannot be limited, or destroyed. …Some say this Atman is slain, and others call It the slayer: they know nothing. How can It slay or who shall slay It? Know this Atman unborn, undying, never ceasing, never beginning, deathless, birthless, unchanging for ever. How can It die the death of the body? Knowing It birthless, knowing It deathless, knowing It endless, for ever unchanging, dream not you do the deed of the killer….”