In Section (2.2), the equation

arose in connection with the study of matrix stability. The matrix *A* ϵ *M*_{n} is given and the right-hand side *H* ϵ *M*_{n} is Hermitian. The unknown matrix *X* ϵ *M*_{n} is to be determined if possible. We are interested in the solvability of (4.0.1.1) in terms of properties of *A* and/or *H*.

It is important to note that the left-hand side of the matrix equation (4.0.1.1) is linear in the unknown *X*. This linearity may be thought of in two ways: If we define the mapping *L*: *M*_{n} → *M*_{n} by *L*(*X*) ≡ *XA* + *A*^{*}*X*, then *L*(*αX + βY*) = *αL*(*X*) + *βL*

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