Hermitian solutions of the equation X = Q + NX−1N* Original Research Article

We consider the matrix equation X = Q + NX−1N*. Its Hermitian solutions are parametrized in terms of the generalized Lagrangian eigenspaces of a certain matrix pencil. We show that the equation admits both a largest and a smallest solution. The largest solution corresponds to the unique positive definite solution. The smallest solution is the unique negative definite solution if and only if N is nonsingular. If N is singular, no negative definite solution exists. An interesting relation between the given equation and a standard algebraic Riccati equation of Kalman filtering theory is also obtained. Finally, we present an algorithm which converges to the positive definite solution for a wide range of initial conditions.