Page 65 - Textos de Matemática Vol. 39

(II) (III) (IV)
(V) (VI)
π
ν
π
ν π+ν
(I)
For i = 1,2,3, and j = 1,2, there exist ni ×ni Hermitian matrices Hi andnj×nj+1 matricesHj,j+1 suchthatIn(Hi)=(πi,νi,∗),rj,j+1≤ rank Hj,j+1 ≤ Rj,j+1 and
⎡⎣ H1 H12 0 ⎤⎦ H= H1∗2 H2 H23
0 H2∗3 H3
has inertia (π, ν, ∗).
Let k ∈ {1, 2, 3}. Let Wkk be any fixed nk × nk Hermitian matrix with inertia(πk,νk,∗).(I)holdswithHk =Wkk.
Let k ∈ {1, 2}. Let Wk,k+1 be any fixed nj × nj+1 matrix with rk,k+1 ≤ rankWk,k+1 ≤ Rk,k+1. (I) holds with Hk,k+1 = Wk,k+1.
For k = 1, 2, 3 let Wkk be any fixed nk × nk Hermitian matrix with inertia (πk,νk,∗). (I) holds with H1 = W11, H2 = W22 and H3 = W33. Let (i,j,k) = (1,2,3) or (2,3,1). Let Wkk be any fixed nk ×nk Hermit- ian matrix with inertia (πk,νk,∗) and let Wij be any fixed ni×nj matrix with rij ≤ rankWij ≤ Rij. (I) holds with Hk = Wkk and Hij = Wij. The following inequalities hold:
≥max{π2,r12 −ν2,r23 −ν2,π1 −ν2 +r23 −R12,π1 −ν3 +r23, π3 −ν1 +r12,π3 −ν2 +r12 −R23,π1 +π2 −R12,
π1 +π3,π2 +π3 −R23,π1 +π2 +π3 −R12 −R23 } ,
≥max{ν2,r12 −π2,r23 −π2,ν1 −π2 +r23 −R12,ν1 −π3 +r23, ν3 −π1 +r12,ν3 −π2 +r12 −R23,ν1 +ν2 −R12,
ν1 +ν3,ν2 +ν3 −R23,ν1 +ν2 +ν3 −R12 −R23 } ,
≤ min { n1 +π2 +n3,π1 +π2 +π3 +R12 +R23,
π1 +π2 +n3 +R12,π1 +n2 +π3,n1 +π2 +π3 +R23 } ,
≤ min { n1 +ν2 +n3,ν1 +ν2 +ν3 +R12 +R23,
ν1 +ν2 +n3 +R12,ν1 +n2 +ν3,n1 +ν2 +ν3 +R23 } ,
≥max{π1 +ν1 +π2 +ν2 −R12,π2 +ν2 +π3 +ν3 −R23,
π1 +ν1 +π2 +ν2 +π3 +ν3 −R12 −R23, π1+ν1−π2−ν2+2r23−R12,π3+ν3−π2−ν2+2r12−R23 } ,
INERTIA OF HERMITIAN MATRICES: AN OPEN PROBLEM 55
Theorem 3.2 ([11]). Let us consider nonnegative integers ni, πi, νi such that πi + νi ≤ ni, for i = 1,2,3, and let 0 ≤ ri,i+1 ≤ Ri,i+1 ≤ min{ni,ni+1}, for i = 1, 2. Then the following conditions are equivalent: