A simple proof for a quadratic inequality

Abstract

The aim of this note is to give a simple proof for a generalization of the following inequality of S. Saitoh [2, Corollary 1.1]:

Let A ₁,..., A _m be N x N positive definite Hermitian matrices. Then for all x ₁,..., x _n in ${\mathbb{C}^N}$ we have the inequality

$${\left( {\sum\limits_{j = 1}^m {{x_j}} } \right)^*}{\left( {\sum\limits_{j = 1}^m {A_j^{ - 1}} } \right)^{ - 1}}\left( {\sum\limits_{j = 1}^m {{x_j}} } \right) \leqslant \sum\limits_{j = 1}^m {x_j^*{A_j}{x_j},} $$

(1)

where * stand for conjugate transpose.