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We propose a technique for calculating and understanding the eigenvaluedistribution of sums of random matrices from the known distribution of thesummands. The exact problem is formidably hard. One extreme approximation tothe true density amounts to classical probability, in which the matrices areassumed to commute; the other extreme is related to free probability, in whichthe eigenvectors are assumed to be in generic positions and sufficiently large.In practice, free probability theory can give a good approximation of thedensity.  We develop a technique based on eigenvector localization/delocalization thatworks very well for important problems of interest where free probability isnot sufficient, but certain uniformity properties apply. Thelocalization/delocalization property appears in a convex combination parameterthat notably, is independent of any eigenvalue properties and yields accurateeigenvalue density approximations.  We demonstrate this technique on a number of examples as well as discuss amore general technique when the uniformity properties fail to apply.

Eigenvalue approximation of sums of Hermitian matrices from eigenvector localization/delocalization