Resultants and Lyapunov matrix equations

Abstract Let B and Q be real n × n matrices. It is well known that the discrete Lyapunov matrix equation Y − B T YB = Q has a unique solution Y if and only if the resultant R(g(x), x n g (1/ x )) is nonzero, where g(x) is the characteristic polynomial of B . Here, by reducing the underlying matrix, we obtain the factorization R(g(x), x n g( 1 x )) = s n t n ( det Θ) 2 where s n and t n are respectively the sum and alternating sum of the coefficients in g(x) . The factor s n t n det Θ is identified with the determinant of a matrix Φ arising in work of R. A. Smith, thereby answering a question raised by N. J. Young. The connection with the analogous theory arising from the continuous Lyapunov matrix equation is established.