Conference on Matrix Analysis and Its Applications, Coimbra, Portugal, 7 - 11 September 2015, pp.187
Abstract
Let A = (Aij) and B = (Bij) be qxq block positive denite matrices in
which each block is an nxn matrix with complex entries. Denote the block-
Hadamard product of A and B by AB which was dened by Horn, Mathias,
and Nakamura. We rst prove some inequalities for the inverse of block
Hadamard product of two block commuting positive denite matrices. For
any C and D of size qxq block matrices under strong commutation assumtions
(C D)(A B)1(C D) (CA1C) (DB1D) (1)
In particular
(A B)1 A1 B1 (2)
Then we give three inequalities which are releated to the Schur complements
of block Hadamard product A B and its inverse (A B)1 as
(A B)= A= B=; (3)
(A B)1= [(A B)=]1 (A=)1 (B=)1; (4)
and
(A B)1= (A1=) (B1=) (A=)1 (B=)1: (5)
Keywords : Block Hadamard product, Schur complements, matrix inequalities.