Resultants and Lyapunov matrix equations (Q909748)

Let B and Q be real \(n\times n\) matrices. Then the discrete Lyapunov matrix equation \(Y-B^ TYB=Q\) has a unique solution Y if and only if the resultant \(R(g(x),x^ ng(1/x))\) is non-zero where g(x) is the characteristic polynomial of B. The author obtains the factorization \(R(g(x),x^ 2g(1/x))=s_ nt_ n(\det \Theta)^ 2\) where \(s_ n\) and \(t_ n\) are respectively the sum and alternating sum of the coefficients in g(x). The factor \(s_ nt_ n\det \Theta\) is identified with the determinant of a matrix \(\Theta\) by R. A. Smith. Thus, the author answers a question raised by R. A. Smith for the equation \(Y-B^ TYB=Q\) to have a unique solution Y when B is a real companion matrix. He also establishes the connection with the analogous theory arising from the continuous Lyapunov matrix equation.