Page 63 - Textos de Matemática Vol. 39

INERTIA OF HERMITIAN MATRICES: AN OPEN PROBLEM 53
Theorem 2.6 ([3]). Let us consider nonnegative integers n, ni, π, ν, πi, νi such thatπi+νi ≤ni,fori=1,2,andlet0≤r≤R≤min{n1,n2}andπ+ν≤ n1 + n2. Then the following conditions are equivalent:
(I) There exist Hermitian matrices Hi, ni × ni, for i = 1, 2, and a matrix H12,n1×n2,suchthatIn(Hi)=(πi,νi,∗),r≤rankH12 ≤Rand
H=H1 H12 H1∗2 H2
has inertia (π, ν, ∗).
(II) Let k ∈ {1, 2}. Let Wkk be any fixed nk × nk Hermitian matrix with
inertia(πk,νk,∗).(I)holdswithHk =Wkk.
(III) Let W be any fixed n1 ×n2 matrix with r ≤ rankW ≤ R. (I) holds with
H12 =W.
(IV) For k = 1, 2 let Wkk be any fixed nk × nk Hermitian matrix with inertia
(πk,νk,∗). (I) holds with H1 = W11 and H2 = W22.
(V) The following inequalities hold:
max{π1,π2,r−ν1,r−ν2,π1+π2−R}≤π≤min{n1+π2,π1+n2,π1+π2+R}, max{ν1,ν2,r−π1,r−π2,ν1+ν2−R} ≤ ν ≤ min{n1+ν2,ν1+n2,ν1+ν2+R},
−ν1 − ν2 ≤ π − ν ≤ π1 + π2 ,
π1 + ν1 + π2 + ν2 − R ≤ π + ν ≤ min {π1 + ν1 + n2 + R, n1 + π2 + ν2 + R} .
Notice that the set of linear inequalities in each theorem is self-πν-dual in the sense that it remains invariant after we transform each inequality into its πν-dual, i.e., the set is the same after the substitution in each inequality of thesymbolsν,π,νi,πi forπ,ν,πi,νi,respectively.
Finally we mention an application of the last theorem. A (symmetric) matrix whose entries are from the set {+,−,0} is called a (symmetric) sign pattern matrix. Sign patterns have become a very popular field of interest in matrix theory (cf. [13] and references therein). For example, from Theorem 2.6, any symmetric matrix with the following (symmetric) sign pattern
⎡⎢ + 0 + − ⎤⎥ A = ⎢⎣ 0 + − + ⎥⎦
+−−0 −+0−
has inertia (2, 2, 0). This means that the inertia of any symmetric matrix with a pattern of signs as A is (2, 2, 0). In this situation we say A requires unique inertia and is sign nonsingular, since there are no zero eigenvalues.