Page 26 - Textos de Matemática Vol. 44

16 A. KOVACˇEC
If the eigenvalues of A do not interlace and the eigenvalues of C do not interlace, then we have four possibilities:
αn >α1 cr >cr+1
(iii)
Note that we have (αk−αl)(ck′ −cl′ )>0 in cases (i), (iv) and (αk−αl)(ck′ −cl′ )<0
incases(ii),(iii),wherealways1≤k,k′ ≤r,r+1≤l,l′ ≤n.
Theorem. Under these conditions, if the eigenvalues of A do not interlace and
the eigenvalues of C do not interlace then WCJ(A) is an interval as follows:
 ]−∞, ciαi] iff (i) or (iv) hold W CJ ( A ) =
 [ ri=1 ciαr−i+1 + ni=r+1 ciαn+r−i+1, ∞[ iff (ii) or (iii) hold
The proof makes necessary the use of a number of lemmas; one of these is the Marcus-Filipenko-type lemma that relates corners z = tr(CU−1AU) of WCJ(A) to a block structure of A and was proved in [BLdPS04] reported on below. As corollaries to the main theorem, one obtains with much ease the following result (a) (by special choices of C) and from this then (b), (c). We always assume A, J, α1 , ..., etc. as in the theorem:
(a) An indefinite version of Ky Fan’s Maximum principle:
Ifα1 <αn and1≤k≤r,then kj=1x∗jJAxj ≤ kj=1αj forallxj ∈Cn such that x∗j Jxl = δjl and conversely. Equality holds if the xj are chosen to be J-orthonormal eigenvalues corresponding to the k greatest eigenvalues.
(b) An indefinite version of the Rayleigh-Ritz theorem:
If α1 < αn, then α1 = maxx∗Jx=1 x∗JAx and αn = minx∗Jx=−1 −x∗JAx and conversely.
(c) An analogue to Schur’s theorem relating the eigenvalues and the diagonal entries of a Hermitian matrix is as follows:
Let a′11 ≥ ··· ≥ a′rr and a′r+1,r+1 ≥ ··· ≥ a′nn be the decreasing rear- rangements of the diagonal entries a11,...,arr and ar+1,r+1,...,ann, re- spectively. If α1 < αn, then kj=1 a′jj ≤ kj=1 αj for all k, 1 ≤ k ≤ n, with equality for k = n.
Similar results hold in the other cases (i), (ii), (iii) defined by inequalities betweentheα1,αr,αn and1,k,r,n.
αr >αr+1 cr >cr+1 (i)
αr >αr+1 cn >c1 (ii)
αn >α1 cn >c1 (iv)