Simultaneous congruence of convex compact sets of Hermitian matrices with constant rank (Q1092138)

Let a compact, convex set S of \(n\times n\) Hermitian matrices be given, and let a fixed n-tuple of signs (plus or minus) be given. Then the authors show that a fixed conjunctive transformation of every member of S exists such that the diagonal of all the resulting matrices has the fixed sign pattern. This theorem requires that every member of S be nonsingular and have exactly one negative eigenvalue. Several other similarly structured theorems are also proved.