A new algorithm for solving the Lyapunov matrix equationX − A*XA = Qis proposed. The method proceeds by reducing to a special case for which an explicit formula is given. The technique is purely algebraic (i.e. involves no iteration), but does not involve the calculation of the characteristic polynomial ofAor reduction to a canonical form. IfQis symmetric and the matrices are of type n × n, the number of multiplications and divisions required is about 4n4. Two simple devices are given whereby the technique can be extended to a wider class of linear matrix equations.