Page 66 - Textos de Matemática Vol. 39

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CARLOS M. DA FONSECA
π+ν
π−ν ν−π
≤min{n1 +n2 +n3,π1 +ν1 +n2 +n3 +R12,
n1 +π2 +ν2 +n3 +R12 +R23,n1 +n2 +π3 +ν3 +R23, π1 +ν1 +π2 +ν2 +n3 +2R12 +R23,
π1 +ν1 +n2 +π3 +ν3 +R12 +R23,
n1 +π2 +ν2 +π3 +ν3 +R12 +2R23 } ,
≤min{π1+π2+π3,π1+π2+π3−ν1+R12,π1+π2+π3−ν3+R23 }, ≤min{ν1+ν2+ν3,ν1+ν2+ν3−π1+R12,ν1+ν2+ν3−π3+R23 } .
With a routine induction argument, based on the partitions developed in [4] or Theorem 3.1 of [11], after an analogous elimination process of redundant inequalities, the author extended Theorem 3.2 to any tridiagonal block decom- position. This result was then applied to symmetric tridiagonal sign patterns.
Shortly thereafter, the author considered a different problem. The aim was the characterization of 3 × 3 partitioned Hermitian matrices (1.1) with zero main diagonal blocks.
Theorem 3.3 ([12]). For i = 1, 2, 3, let ni be a nonnegative integer, and for i < j ≤ 3, let 0 ≤ rij ≤ Rij ≤ min{ni,nj}. Then the following conditions are equivalent:
(I) Fori=1,2,andfori<j≤3,thereexistni×nj matricesHij such that rij ≤ rank Hij ≤ Rij and
⎡ 0 H12 H13 ⎤ H=⎣H1∗2 0 H23⎦
H1∗3 H2∗3 0
has inertia (π, ν, ∗).
(II) Let (i,j) ∈ {(1,2),(1,3),(2,3)} and let Wij be any fixed ni ×nj matrix
with rij ≤ rankWij ≤ Rij. (I) holds with Hij = Wij.
(III) The following inequalities hold:
max{r12,r13,r23}≤π,ν ≤min{n1 +R23,n2 +R13,n3 +R12} ,
π−ν,ν−π≤min{R12,R13,R23} , 2π − ν, 2ν − π ≤ R12 + R13 + R23 , π + ν ≤ n1 + n2 + n3 .