For the (b)-case:
- (i) note that \(\hat{\boldsymbol{f}}_1(k) = \boldsymbol{S}_1 (\boldsymbol{y}-\hat{\boldsymbol{f}}_2(k-1))\) and\(\hat{\boldsymbol{f}}_2(k) = \boldsymbol{S}_2 (\boldsymbol{y}-\hat{\boldsymbol{f}}_1(k-1))\) at iteration \(k\),
- (ii) unroll the recursive relationship to express \(\hat{\boldsymbol{f}}_i(k)\) as a function of \(\hat{\boldsymbol{f}}_i(0)\),
- (iii) geometric matrix series.