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THE INERTIA OF HERMITIAN MATRICES
WITH A PRESCRIBED 3 × 3 BLOCK DECOMPOSITION:
AN OPEN PROBLEM
CARLOS M. DA FONSECA
Dedicated to Professor E. Marques de Sa´ on the occasion of his 60th anniversary.
Abstract. Given nonnegative integers n1, n2, n3, we consider partitioned Hermitian matrices H of the form
⎡ H1 H = ⎣ H 1∗ 2
H12 H 2
H13 ⎤ H 2 3 ⎦
H1∗3
{In(H)|In(Hi) = (πi,νi,δi) and rij ≤ rankHij ≤ Rij, for 1 ≤ i < j ≤ 3}
H3
where each block Hij is ni × nj . The characterization of the set of inertias
by a system of linear inequalities involving ni , πi , νi , δi , rij and Rij is still an open problem. This is a survey note based on S´a’s inertia papers devoted to this problem. Some recent developments are provided and a conjecture is established.
1. Introduction
Define the inertia of an n × n Hermitian matrix H, as the ordered triple
In(H) = (π,ν,δ)
where π is the number of positive eigenvalues of A, ν is the number of negative eigenvalues of A, and δ is the number of zero eigenvalues of A, all counting multiplicities, with δ = n − π − ν. Sometimes it is also denoted by (π, ν, ∗) or simply by (π, ν).
The characterization of the set of inertias having prescribed complemen- tary principal submatrices have been studied since the 1960’s, and there are still some open questions. Given a Hermitian matrix H, we consider the following partition
⎡⎣H1 H12 H13⎤⎦ (1.1) H = H1∗2 H2 H23 .
H1∗3 H2∗3 H3
2000 Mathematics Subject Classification. 15A18, 15A42, 15A57.
Key words and phrases. Inertia, Hermitian matrices, block decomposition.
This work was supported by CMUC - Centro de Matem´atica da Universidade de Coimbra.
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H2∗3