Page 61 - Textos de Matemática Vol. 39

INERTIA OF HERMITIAN MATRICES: AN OPEN PROBLEM 51
One of the most interesting aspects of this fundamental theorem is the reduction of the original problem to the identical one for diagonal matrices. Marques de Sa´ also included a generalization of this result for a finite sum of Hermitian matrices.
At the same time, Cain and Marques de Sa´ ([2]) considered for the first time the completion problem of characterizing the set of inertias
{In(H)|In(Hi) = (πi,νi,δi), for i = 1,2}, (2.1) H = H1 X
for the partition
in terms of the inertia of the complementary diagonal blocks.
X∗ H2
Theorem 2.3 ([2]). Let us consider nonnegative integers n, π, ν, ni, πi, νi such that πi +νi ≤ ni, for i = 1,2, and π+ν ≤ n. Then the following conditions are equivalent:
(I) Given Hermitian matrices Hi, ni × ni, such that In(Hi) = (πi, νi, ∗), there exists an n1 × n2 matrix X such that In (H) = (π, ν, ∗).
(II) For i = 1,2, there exist Hermitian matrices Hi, ni ×ni, such that In(Hi) = (πi,νi,∗), and an n1 × n2 matrix X such that In(H) = (π, ν, ∗).
(III) The following inequalities hold:
max {π1, π2} ≤ π ≤ min {n1 + π2, π1 + n2} ,
max {ν1, ν2} ≤ ν ≤ min {n1 + ν2, ν1 + n2} , −ν1 − ν2 ≤ π − ν ≤ π1 + π2 .
This result was extended to any partition in [1] and, later, in [6]. In fact Cain characterized the inertias of Hermitian matrices with m-by-m (m > 2) block decompositions in terms of a system of linear inequalities involving the orders of the blocks and the inertias of the main diagonal blocks.
Given the integers x1,...,xm and y1,...,ym, let us define the set
L k ( m , x ∗ , y ∗ ) = m i n ⎧⎨⎩ x i + y i | I ⊂ { 1 , . . . , m } a n d | I | = k ⎫⎬⎭ ,
i̸∈I i∈I
for k = 1,...,m.
Theorem 2.4 ([1]). Let π, ν, m, ni, πi, νi be nonnegative integers and πi+νi ≤ni,fori=1,...,m,andsetn=n1+···+nm withπ+ν≤n. The following conditions are equivalent:
(I) There exists an n×n Hermitian matrix H = [Hij ] where Hij are ni ×nj blocks, satisfying In(Hii) = (πi, νi, ∗) and In(H) = (π, ν, ∗).