1 The eigenvalue distribution function For an N × N matrix AN , the eigenvalue distribution function 1 (e.d.f.) F AN (x) is defined as F AN (x) = Number of eigenvalues of AN ≤ x . (1) N As defined, the e.d.f. is right continuous and possibly atomic i.e. with step discontinuities at discrete points. In practical terms, the derivative of (1), referred to as the (eigenvalue) level density, is simply the
Interpretation An interpretation I of F is , where D is a non-empty set called the domain of individuals. I0 is a mapping defined on the constants of F satisfying 1. If c is an individual constant, then I0(c) ∈ D. 2. If f n is an n-ary function constant, then I0(f n) : Dn → D