Page 127 - Textos de Matemática Vol. 39

PARTIAL SPECTRA OF HERMITIAN SUMS 117
3. The case s=1
Here we are interested, given k, in the possible numbers γk occurring as the k-th eigenvalue of sums A + B, A and B Hermitian with the given spectra α and β.
The following inequalities concerning a single eigenvalue of A + B have been known for a long time, and associated with the name of Weyl:
γi+j−1 ≤αi +βj , γi+j−n ≥αi +βj
(for all admissible values of the indices). The first is easily proved from the extremal characterizations of eigenvalues of Hermitian matrices using the as- sociated quadratic forms, and the second follows from the first applied to −A and −B.
We rewrite Weyl’s inequalities as follows:
αi +βk−i+n ≤ γk (i=k,...,n) and γk ≤ αi +βk−i+1 (i=1,...,k) or
maxk≤i≤n αi + βk−i+n ≤ γk ≤ min1≤i≤k αi + βk−i+1 . Theorem 3.1 ([6]; see also [11, 14]). These conditions are sufficient.
Proof. The set of possible γ is an interval. By the necessity, that interval is contained in
[ maxk≤i≤n αi + βk−i+n , min1≤i≤k αi + βk−i+1 ] .
The lower bound is attained by the k-th eigenvalue of A + B, where
A = diag(α1,...,αn) , B = diag(βk,βk−1,...,β2,β1,βk+1,βk+2,...,βn).
The upper bound is attained by the k-th eigenvalue of A + B, where
A = diag(α1,...,αn) , B = diag(β1,β2,...,βk−1,βn,βn−1,...,βk+1,βk).
Two comments are in order concerning this result. First, the extreme points of the realizable set are produced with diagonal matrices. Second, the rel- evant inequalities are precisely those appearing in Horn’s list as explicit bounds on a single eigenvalue: the first family of Weyl inequalities, γi+j−1≤αi+βj, is clearly the list of 1-term inequalities appearing there; the second family, γi+j−n ≥ αi + βj , consists exactly of the inequalities obtained from the trace condition together with the (n−1)-term inequalities in the Horn list [4], since if I has length n − 1, the (n − 1)-tuple λ(I ) has only 1’s and 0’s, and in that situ- ation it is not difficult to find all possible cases in which λ(K) ∈ E[λ(I), λ(J)].