Inertia of the Stein transformation with respect to some nonderogatory matrices (Q1923148)

The Stein transformation is \(S_A(K) = K-AKA^*\) where \(K\) is Hermitian. The author considers the relation between the inertia of \(K\) and of \(S_A(K)\) with respect to the unit circle, that is, counts of eigenvalues inside, outside, and on the unit circle. For instance, if \(A\) has distinct eigenvalues all of which are on the unit circle, \(K\) has no eigenvalues on the unit circle, then we can realize all possibilities when \(S_A(K)\) has at least one eigenvalue inside and at least one eigenvalue outside the unit circle (Theorem 1). The Stein transformation in other settings is related to Lyapunov stability.