Page 69 - Textos de Matemática Vol. 39

INERTIA OF HERMITIAN MATRICES: AN OPEN PROBLEM 59
π+ν ≤min{n1+n2+n3,
π1 +ν1 +n2 +n3 +R12 +R13,
n1 +π2 +ν2 +n3 +R12 +R23,
n1 +n2 +π3 +ν3 +R13 +R23,
π1 +ν1 +π2 +ν2 +n3 +2R12 +R13 +R23, π1 +ν1 +n2 +π3 +ν3 +R12 +2R13 +R23, n1 +π2 +ν2 +π3 +ν3 +R12 +R13 +2R23 }
π−ν ≤min{n1 +π2 +π3,π1 +n2 +π3,π1 +π2 +n3,
π1 +π2 +π3 +R12,π1 +π2 +π3 +R13,π1 +π2 +π3 +R23,
π1 +π2 +π3 −ν1 +R12 +R13, π1 +π2 +π3 −ν2 +R12 +R23, π1 + π2 + π3 − ν3 + R13 + R23 } ,
ν−π ≤min{n1 +ν2 +ν3,ν1 +n2 +ν3,ν1 +ν2 +n3,
ν1 +ν2 +ν3 +R12,ν1 +ν2 +ν3 +R13,ν1 +ν2 +ν3 +R23,
ν1 +ν2 +ν3 −π1 +R12 +R13,
ν1 +ν2 +ν3 −π2 +R12 +R23,
ν1 + ν2 + ν3 − π3 + R13 + R23 } ,
−R12 −R13 −R23 ≤ π−2ν ≤π1 +π2 +π3, −R12 −R13 −R23 ≤ ν−2π ≤ν1 +ν2 +ν3 .
It is not difficult to generalize Theorem 4.1 to m×m block decompositions, but it is not clear how to generalize Conjecture 4.2 to the m × m case. Cain’s result, Theorem 2.4, suggests new inequalities involving
π − 3ν, ν − 3π, . . . , π − (m − 1)ν, ν − (m − 1)π. References
[1] B.E. Cain, The inertia of a Hermitian matrix having prescribed diagonal blocks, Linear Algebra Appl. 37 (1981), 173 − 180.
[2] B.E. Cain and E. Marques de S´a, The inertia of a Hermitian matrix having prescribed complementary principal submatrices, Linear Algebra Appl. 37 (1981), 161 − 171.
[3] B.E. Cain and E. Marques de S´a, The inertia of Hermitian matrices with a prescribed 2 × 2 block decomposition, Linear Multilinear Algebra 31 (1992), 119 − 130.
[4] B.E. Cain and E. Marques de S´a, The inertia of certain skew-triangular block matrices, Linear Algebra Appl. 160 (1992), 75 − 85.
[5] D. Carlson and T. Markham, Schur complements of diagonally dominant matrices, Czechoslovak Math. J. 29(104) (1979), 246 − 251.
[6] N. Cohen and J. Dancis, Inertias of block band matrix completions, SIAM J. Matrix Anal. Appl. 19 (1998), 583 − 612.