Presents a new approach to the Chandrasekhar recursions and some generalizations thereof. The derivation uses the generalized Schur recursions, which are O(N/sup 2/) recursions for the triangular factorization of N/spl times/N matrices having a certain Toeplitz-like displacement structure. It is shown that when the extra structure provided by an underlying state-space model is properly incorporated into the generalized Schur algorithm, it reduces to the Chandrasekhar recursions, which are O(Nn/sup 2/) recursions for estimating the n-dimensional state of a time-invariant (or constant-parameter) system from N measured outputs. It is further noted that the generalized Schur algorithm factors more general structured matrices, and this fact is readily used to extend the Chandrasekhar recursions to a class of time-variant state-space models, special cases of which often arise in adaptive filtering.