<P> Referring back to our expanded formula for the a posteriori error covariance, </P> <Dl> <Dd> P k ∣ k = P k ∣ k − 1 − K k H k P k ∣ k − 1 − P k ∣ k − 1 H k T K k T + K k S k K k T (\ displaystyle \ mathbf (P) _ (k \ mid k) = \ mathbf (P) _ (k \ mid k - 1) - \ mathbf (K) _ (k) \ mathbf (H) _ (k) \ mathbf (P) _ (k \ mid k - 1) - \ mathbf (P) _ (k \ mid k - 1) \ mathbf (H) _ (k) ^ (\ mathrm (T)) \ mathbf (K) _ (k) ^ (\ mathrm (T)) + \ mathbf (K) _ (k) \ mathbf (S) _ (k) \ mathbf (K) _ (k) ^ (\ mathrm (T))) </Dd> </Dl> <Dd> P k ∣ k = P k ∣ k − 1 − K k H k P k ∣ k − 1 − P k ∣ k − 1 H k T K k T + K k S k K k T (\ displaystyle \ mathbf (P) _ (k \ mid k) = \ mathbf (P) _ (k \ mid k - 1) - \ mathbf (K) _ (k) \ mathbf (H) _ (k) \ mathbf (P) _ (k \ mid k - 1) - \ mathbf (P) _ (k \ mid k - 1) \ mathbf (H) _ (k) ^ (\ mathrm (T)) \ mathbf (K) _ (k) ^ (\ mathrm (T)) + \ mathbf (K) _ (k) \ mathbf (S) _ (k) \ mathbf (K) _ (k) ^ (\ mathrm (T))) </Dd> <P> we find the last two terms cancel out, giving </P>

Rd case of one step forward two steps back