Solvability of linear matrix equations in a symmetric matrix variable

We study the solvability of generalized linear matrix equations of the Lyapunov type in which the number of terms involving products of the problem data with the matrix variable can be arbitrary. We show that contrary to what happens with standard Lyapunov equations, which have only two terms, these generalized matrix equations can have unique solutions but the associated matrix representation in terms of Kronecker products can be singular. We show how a simple modification to the equation can lead to a matrix representation that does not suffer from this deficiency.