Linear backfitting equations ($f_j+T_j{\sum }_{k\ne j}{f}_k=T_{j}Cy$)

For linear smoothers, let $C=I-11^T/n$ and $T_j=C{S}_{j}C$. The centered fixed-point equations of the backfitting algorithm have diagonal identity blocks and off-diagonal blocks $T_j$. On the centered component space, a consistent system has a unique answer exactly when its homogeneous system has only the zero solution. For two components the reduced equation is $(I-T_1{T}_2)f_1=T_1(I-T_2)Cy$. A common centered vector reproduced by both smoothers yields opposite perturbations of the components without changing their sum.

// Target: probability-and-statistics.bigb