ON SIMULTANEOUS CONGRUENCE AND NORMS OF HERMITIAN MATRICES

Let A1, ..., A(m), B1, ..., B(m) be n x n complex Hermitian matrices. It is said that B1, ..., B(m) are simultaneously congruent to A1, ..., A(m) if there exists an invertible S such that S* A(i)S = B(i), i = 1, ..., m. In this paper, inf parallel-to I - S parallel-to, 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 parallel-to I - S parallel-to is of the same magnitude as that of parallel-to B1 - A1 parallel-to + ... + parallel-to B(m) - A(m) parallel-to. 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.