Discrete Riccati equation for systems with singular transition matrix

We consider the discrete algebraic Riccati equation with singular coefficient matrix in relation to the linear quadratic regulator problem. It is shown that the positive definite solution of the Riccati equation can be obtained via a similar equation of reduced size. We derive the result by manipulating the performance index of the associated optimal regulator problem. This is an extension to a similar result in which some restriction applies to the form of the weighting matrix in the performance index.