Page 67 - Textos de Matemática Vol. 39

INERTIA OF HERMITIAN MATRICES: AN OPEN PROBLEM 57
Let us go back to sign patterns. According to Theorem 3.3, the possible inertias for the symmetric sign pattern
⎡000+++⎤ ⎢ 0 0 0 − + 0 ⎥ ⎢ 0 0 0 − + − ⎥ ⎢⎣ + − − 0 0 + ⎥⎦ +++00+ +0−++0
can be easily calculated: (2, 2, 2) (2, 3, 1), (3, 2, 1), and (3, 3, 0). 4. A conjecture
So far, the main theorems state a number of equivalent conditions. In each theorem the last condition is a large collection of inequalities. While the characterization of the set of inertias for 2 × 2 block decompositions is solved, the case 3 × 3 seems to be far from decided.
Although we are not certain what the general theorem for 3 × 3 block decompositions should say, here is a partial result without that long and last complicated condition:
Theorem 4.1 ([14]). Let us assume that the quantities πi , νi , ni , for i = 1,2,3, are nonnegative and πi+νi ≤ni, and 0≤rij ≤Rij ≤min{ni,nj}, 1 ≤ i < j ≤ 3. Then the following conditions are equivalent:
(I) For i = 1,2,3, and j = 2,3, there exist ni ×ni Hermitian matrices Hi and ni×nj matrices Xij such that In(Hi) = (πi,νi,∗), rij ≤ rankXij ≤ Rij when i<j and
⎡⎣H1 X12 X13⎤⎦ H= X1∗2 H2 X23
X1∗3 X2∗3 H3
has inertia (π, ν, ∗).
(II) Let k ∈ {1, 2, 3}. Let Wkk be any fixed Hermitian matrix of order nk
with inertia (πk , νk , ∗). (I) holds with Hk = Wkk .
(III) Let j,k ∈ {1,2,3} such that j < k. Let Wjk be any fixed nj ×nk matrix
with rjk ≤ rankWjk ≤ Rjk. (I) holds with Xjk = Wjk.
(IV) 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.
(V) Let (i,j,k) = (1,2,3), (1,3,2), or (2,3,1). Let Wkk be any fixed
nk × nk Hermitian matrix with inertia (πk , νk , ∗) and let Wij be any fixed ni ×nj matrix with rij ≤ rank Wij ≤ Rij . (I) holds with Hk = Wkk
andXij =Wij.