On the stability of recursive least squares in the Gauss-Markov model

In the Gauss-Markov regression model, one can always update the least square estimate of the slope vector, given new observations at the values of the explanatory variables. The updated estimate is often considered as a time-varying state of an auto-regressive system in Kalman filtering and recursive least squares theory. This note shows that the auto-regressive matrix of this dynamic system once centered has its largest eigenvalues equal to 1 and one eigenvalue that is less than 1