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A new method is presented for the numerical deterruination of the solution of the steady-state matrix Riccati equation. The equation is converted to a canonical form corresponding to Luenberger's canonical representation for controllable multivariable systems. Three special matrices closely associated with Luenberger's canonical form are defined and two related lemmas are established. These results are used to obtain concise expressions for the eigenvectors of the Hamiltonian matrix associated with the canonical Riccati equation in terms of the solutions of a much simpler reduced Hamiltonian system. Using a theorem due to Potter the solution of the Riccati equation is written in terms of the concise eigenvector expressions. The method is particularly well suited to problems in which the ratio of system states to system inputs is large and it can lead to a 26 to 1 reduction in the computational effort required to solve the Riccati equation.