Matter-energy conversion was an incredibly powerful and dangerous application of magic in which the magician converts a portion of the matter making up their body, or presumably the matter making up any other object, into pure energy, which erupts outward in a cataclysmic explosion. The only confirmed users of this ability were Thuviel, who sacrificed himself in this manner in a final attempt to defeat the Forsworn on Vroengard, and Galbatorix, who at the end of the Rider War used this power to commit suicide in a final act of defiance after his defeat, causing grave damage to Urû'baen in the process, and completely destroying his citadel.
It is implied that Angela the herbalist may also have known how to use this power, as when Roran asked her if there was anything she could do to stop Lord Barst, she replies that she could, but that if she did it would result in the deaths of her and everyone else in Urû'baen, except for Galbatorix who was too strongly shielded.
Besides the explosion itself, which could only be described as cataclysmic, this use of magic also had a deadly side effect; the release of an "invisible poison" upon the area surrounding the site of the explosion. Given that this ability involves converting matter into energy, it is likely that this "invisible poison" is some form of radiation; the effects it had on the island of Vroengard support this assumption.