The boundedness of the Kalman filter, as the first cornerstone of its stability analysis, has been proved in the classical literature through upper bounds of non-recursive filters in the sense of the trace of the state estimation error covariance. In this paper, an upper bound of the Kalman filter prediction error covariance is established in the sense of matrix positive definiteness, based on a bounded recursive non-optimal filter. The boundedness of the error covariance is a prerequisite for the definition of a Lyapunov function involved in the state estimation error dynamics stability analysis.