Page 62 - Textos de Matemática Vol. 39

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CARLOS M. DA FONSECA
(II) Given Hermitian matrices Xi of order ni, with In(Xi) = (πi, νi, ∗) for i = 1,...,m, there exists an n×n Hermitian matrix H = [Hij], where Hij are ni × nj blocks, satisfying Xi = Hii and In(H) = (π, ν, ∗).
(III) The following inequalities hold:
max{π1,...,πm} ≤ π , max{ν1,...,νm} ≤ ν ,
π−(k−1)ν ≤ Lk(m,n∗,π∗), fork=1,...,m, ν−(k−1)π ≤ Lk(m,n∗,ν∗), fork=1,...,m.
Using mainly Schur complements and Poincar´e inequalities, in [6] Cohen and Dancis also generalized Theorem 2.3 in a different way. Constructing the so-called inertial polygon, they provided a description for the set of inertias of a 3 × 3 block bordered matrix in terms of (seven) linear inequalities involving inertias and ranks of some particular submatrices.
Theorem 2.5 ([6]). Given the block bordered matrix
⎡⎣H1 H12 X⎤⎦ H= H1∗2 H2 H23
X ∗ H 2∗ 3 H 3
where Hi is a Hermitian matrix of order ni , for i = 1, 2, 3, let us define
In H1 H12 =(π ,ν ,∗) and In H2 H23 =(π ,ν ,∗), H1∗2 H2 12 12 H2∗3 H3 23 23
r = rank(H1∗2H2H23),
∆′ = r − rank (H1∗2H2) , ∆′′ = r − rank (H2H23),
π0 =max{π12 +∆′,π23 +∆′′} and ν0 =max{ν12 +∆′,ν23 +∆′′}. Then given nonnegative integers π and ν, there exists an n1 × n3 matrix X
such that In(H) = (π,ν,∗), if and only if π0≤π≤min{n1+π23,n3+π12}, ν0≤ν≤min{n1+ν23,n3+ν12},
r − ν12 − ν23 ≤ π − ν ≤ π12 + π23 − r , π + ν ≤ n1 + n3 + r .
Notice that making n2 = 0 we get Theorem 2.3.
If we impose a restriction on the off-diagonal block in (2.1), then we will realize how much the inertias of the pair of complementary principal submatri- ces and the rank of the off-diagonal block influence the inertia of H. Cain and Marques de Sa´ considered that restriction in [3] and proved the following: