Page 64 - Textos de Matemática Vol. 39

54 CARLOS M. DA FONSECA
3. Hermitian skew-triangular block matrices
In 1992, Cain and Marques de S´a ([4]) extended the methods given by Haynsworth and Ostrowski in [16] for estimating the inertia of 3 × 3 skew- triangular block matrices
⎡⎣H1 H12 H13⎤⎦ (3.1) H= H1∗2 H2 0 .
H1∗3 0 0
In fact they generalize their previous main results (cf. Theorem 2.6) and the
proofs rely on skillful use of the techniques developed in [3, 18].
Theorem 3.1 ([4]). Let us consider nonnegative integers ni,πi,νi such that πi+νi ≤ni,fori=1,2,andlet0≤r1i ≤R1i ≤min{n1,ni},fori=2,3. Then the following conditions are equivalent:
(I) For i = 1, 2, and j = 2, 3, there exist Hermitian matrices Hi, ni × ni, and matrices H1j, n1 × nj, such that In(Hi) = (πi,νi,∗), r1j ≤ rank H1j ≤ R1j and H with the block partition (3.1) has inertia (π, ν, ∗).
(II) Let k ∈ {1, 2}. Let Wkk be any fixed nk × nk Hermitian matrix with inertia(πk,νk,∗).(I)holdswithHk =Wkk.
(III) Let k ∈ {2,3}. Let W1,k be any fixed n1 × nk matrix with r1k ≤ rank W1k ≤ R1k. (I) holds with H1k = W1k.
(IV) For k = 1, 2 let Wkk be any fixed nk × nk Hermitian matrix with inertia (πk,νk,∗). (I) holds with H1 = W11 and H2 = W22.
(V) Let W22 be any fixed n2 × n2 Hermitian matrix with inertia (π2, ν2, ∗) and let W13 be any fixed n1 × n3 matrix with r13 ≤ rank W13 ≤ R13. (I) holds with H2 = W22 and X13 = W13.
(VI) The following inequalities hold:
π ≥ ν ≥ π ≤ ν ≤
π−ν ≤ ν−π ≤ π+ν ≥
max{π1,π2 +r13,π1 +π2 −R12,r12 −ν1,r12 −ν2} , max{ν1,ν2 +r13,ν1 +ν2 −R12,r12 −π1,r12 −π2} , min{n1 +π2,π1 +n2 +R13,π1 +π2 +R12 +R13} , min{n1 +ν2,ν1 +n2 +R13,ν1 +ν2 +R12 +R13} , min{π1 +π2,π1 +π2 −ν2 +R12} ,
min{ν1 +ν2,ν1 +ν2 −π2 +R12} , π1 +π2 +ν1 +ν2 −R12 ,
π+ν ≤ min{n1 +n2 +R13,n1 +π2 +ν2 +R12 +R13, π1 +ν1 +n2 +R12 +2R13} .
Using the same techniques, Theorem 3.1 was improved by the author in [11], having the following block tridiagonal version.