Matrix linear equations ATX+XB=P and ΦTXψ+P=X arise in control theory, and the efficient computer solution of these equations is an important problem. This paper develops an algebraic method of solution of the equations, via a similarity transformation of system matrices to companion form. Solution evaluation by this method is three to five times faster than by standard methods. The method also shows the formal equivalence of state-space and transform expressions for the output covariance of single-input/single-output systems, and provides an improved algorithm for evaluating such terms.