The Linear Quadratic Consensus Control (LQCC) problem is a relaxation of the classical Linear Quadratic Regulation (LQR) problem, that consists of asymptotically driving the state of the system to a "consensus" point in which all coordinates are equal, in such a way that a quadratic cost function on the transient of the state trajectory and the input is minimized. Instead of requiring asymptotic stability as in the case of LQR, the appropriate condition in this case is semi-stability. This paper shows that the solution of the LQCC can be obtained by solving an algebraic matrix system that extends the classical algebraic Riccati equation. Moreover, the solution of this system can be obtained from the eigenvectors of a Hamiltonian matrix in a manner that is an extension of the familiar Schur-based algorithm for solving the Riccati equation. It is possible that the LQCC has an optimal solution even if the corresponding LQR problem does not have one. However, if the LQR problem does have a solution, an interesting connection between the solutions of the two problems can be used to solve the LQCC problem in cases in which the previous approach may not be practial. © 2012 by TNO Organization, The Netherlands. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.