In Section (2.2), the equation
arose in connection with the study of matrix stability. The matrix A ϵ Mn is given and the right-hand side H ϵ Mn is Hermitian. The unknown matrix X ϵ Mn is to be determined if possible. We are interested in the solvability of (18.104.22.168) in terms of properties of A and/or H.
It is important to note that the left-hand side of the matrix equation (22.214.171.124) is linear in the unknown X. This linearity may be thought of in two ways: If we define the mapping L: Mn → Mn by L(X) ≡ XA + A*X, then L(αX + βY) = αL(X) + βL