On Ranges of Lyapunov Transformations. III
Raphael Loewy · SIAM Journal on Applied Mathematics · 1976
The Lyapunov transformation $\mathcal{L}_A $ corresponding to the matrix $A \in \mathbb{C}^{n,n} $ is a linear transformation on the space $\mathcal{H}_n $ of Hermitian matrices $H \in \mathbb{C}^{n,n} $, of the form $\mathcal{L}_A (H) = AH + HA^ * $. Let \[ C_1 (A) = \{ {AH + HA^ * :H\,{\text{is Hermitian and positive semidefinite}}} \}\]. Given $A \in \mathbb{C}^{n,n} $ such that $\mathcal{L}_A $ is invertible, we characterize the matrices $B \in \mathbb{C}^{n,n} $ such that $C_1 (A) = C_1 (B)$.