Linear Maps That Preserve an Inertia Class

Abstract

The purpose of this paper is to prove the following result: Let $n\geqq 3$ and let r, s be given positive integers such that $r \ne s$ and $r + s\leqq n$. Let $\mathcal{H}_n $ denote the space of all $n \times n$ hermitian matrices. Suppose that $T:\mathcal{H}_n \to \mathcal{H}_n $ is a linear transformation that maps the set of all matrices with r positive eigenvalues and s negative eigenvalues into itself. Then there exists an $n \times n$ nonsingular matrix S such that either $T(A) = S^ * AS$ for all $A \in \mathcal{H}_n$ or $T(A) = S^ * A^t S$ for all $A \in \mathcal{H}_n $. This gives an affirmative answer to a problem raised by Johnson and Pierce.