Some Equalities and Inequalities for the Hermitian Moore-Penrose Inverse of Triple Matrix Product with Applications
Received:August 14, 2014  Revised:December 22, 2014
Key Words: Moore-Penrose inverse   reverse-order law   rank   inertia   L\"owner partial ordering  
Fund Project:Supported by the National Natural Science Foundation of China (Grant No.11271384).
Author NameAffiliation
Yongge TIAN China Economics and Management Academy, Central University of Finance and Economics, Beijing 100081, P. R. China 
Wenxing GUO School of Statistics and Mathematics, Central University of Finance and Economics, Beijing 100081, P. R. China 
Hits: 2826
Download times: 2532
Abstract:
      We investigate relationships between the Moore-Penrose inverse $(ABA^{*})^{\dag}$ and the product $[(AB)^{(1,2,3)}]^{*}B(AB)^{(1,2,3)}$ through some rank and inertia formulas for the difference of $(ABA^{*})^{\dag} - [(AB)^{(1,2,3)}]^{*}B(AB)^{(1,2,3)}$, where $B$ is Hermitian matrix and $(AB)^{(1,2,3)}$ is a $\{1,\, 2,\,3\}$-inverse of $AB$. We show that there always exists an $(AB)^{(1,2,3)}$ such that $(ABA^{*})^{\dag}=[(AB)^{(1,2,3)}]^{*}B(AB)^{(1,2,3)}$ holds. In addition, we also establish necessary and sufficient conditions for the two inequalities $(ABA^{*})^{\dag} \succ [(AB)^{(1,2,3)}]^{*}B(AB)^{(1,2,3)}$ and $(ABA^{*})^{\dag} \prec [(AB)^{(1,2,3)}]^{*}B(AB)^{(1,2,3)}$ to hold in the L\"owner partial ordering. Some variations of the equalities and inequalities are also presented. In particular, some equalities and inequalities for the Moore-Penrose inverse of the sum $A + B$ of two Hermitian matrices $A$ and $B$ are established.
Citation:
DOI:10.3770/j.issn:2095-2651.2015.03.010
View Full Text  View/Add Comment