On Simultaneous Congruence and Norms of Hermitian Matrices

Abstract

Let $A_1 , \cdots ,A_m ,B_1 , \cdots ,B_m $ be $n \times n$ complex Hermitian matrices. It is said that $B_1 , \cdots ,B_m $ are simultaneously congruent to $A_1 , \cdots ,A_m $ if there exists an invertible S such that $S^ * A_i S = B_i ,i = 1, \cdots ,m$. In this paper, $\inf \| I - S \|$, as S ranges over all invertible matrices which afford this simultaneous congruence, are studied. If one of the $A_i $ is positive definite, it turns out that the growth of $\inf \| I - S \|$ is of the same magnitude as that of $\| B_1 - A_1 \| + \cdots + \| B_m - A_m \|$. A counterexample with $m = 2$ is given to show that this result can be false if none of the $A_i $’s is positive definite. An analogous result for simultaneous unitary congruence of matrices is also proved.