Honeycombs and Sums of Hermitian Matrices, Volume 48, Number 2

I n 1912 Hermann Weyl [W] posed the following problem: given the eigenvalues of two n× n Hermitian matrices A and B, how does one determine all the possible sets of eigenvalues of the sum A + B? When n = 1, the eigenvalue of A + B is of course just the sum of the eigenvalue of A and the eigenvalue of B, but the answer is more complicated in higher dimensions. Weyl’s partial answers to this problem have since had many direct applications to perturbation theory, quantum measurement theory, and the spectral theory of selfadjoint operators. The purpose of this article is to describe the complete resolution to this problem, based on recent breakthroughs [Kl], [HR], [KT], [KTW]. To standardize the notation, we shall always write the eigenvalues of an n× n Hermitian matrix as a weakly decreasing n-tuple λ = (λ1 ≥ . . . ≥ λn) of real numbers. Thus, for instance, the eigenvalues of diag(3,2,5,3) are (5,3,3,2). To illustrate Weyl’s problem, suppose that n = 2 and that A , B have eigenvalues (3,0) and (5,0) respectively. Then one can easily verify that A + B can have eigenvalues (8,0) or (5,3) or, more generally, (8− a,a) for any 0 ≤ a ≤ 3. This turns out to be the complete set of possibilities; A + B cannot have eigenvalues (9,−1) or (7,0) or (4,4), etc. Let us denote the eigenvalues of A , B, and A + B as λ, μ, and ν respectively; thus λ2 is the second largest eigenvalue of A , etc. It is fairly easy to obtain necessary conditions on the triple λ, μ, ν. For instance, from the simple observation that the trace of A + B must equal the sum of the traces of A and B, we obtain the condition