Some inequalities and Schur complements of block Hadamard product


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Özel M.

Conference on Matrix Analysis and Its Applications, Coimbra, Portugal, 7 - 11 September 2015, pp.187

  • Publication Type: Conference Paper / Summary Text
  • City: Coimbra
  • Country: Portugal
  • Page Numbers: pp.187
  • Dokuz Eylül University Affiliated: Yes

Abstract

Abstract

Let A = (Aij) and B = (Bij) be qxq block positive de nite matrices in

which each block is an nxn matrix with complex entries. Denote the block-

Hadamard product of A and B by A B which was de ned by Horn, Mathias,

and Nakamura. We rst prove some inequalities for the inverse of block

Hadamard product of two block commuting positive de nite matrices. For

any C and D of size qxq block matrices under strong commutation assumtions

(C D)(A B)􀀀1(C D)  (CA􀀀1C) (DB􀀀1D) (1)

In particular

(A B)􀀀1  A􀀀1 B􀀀1 (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=  (A􀀀1= ) (B􀀀1= )  (A= )􀀀1 (B= )􀀀1: (5)

Keywords : Block Hadamard product, Schur complements, matrix inequalities.