<Dd> ∑ n = 0 2 L − 1 (I − A) n = ∏ l = 0 L − 1 (I + (I − A) 2 l) (\ displaystyle \ sum _ (n = 0) ^ (2 ^ (L) - 1) (\ mathbf (I) - \ mathbf (A)) ^ (n) = \ prod _ (l = 0) ^ (L - 1) (\ mathbf (I) + (\ mathbf (I) - \ mathbf (A)) ^ (2 ^ (l)))). </Dd> <P> Therefore, only 2 L − 2 (\ displaystyle 2L - 2) matrix multiplications are needed to compute 2 L (\ displaystyle 2 ^ (L)) terms of the sum . </P> <P> More generally, if A is "near" the invertible matrix X in the sense that </P> <Dl> <Dd> lim n → ∞ (I − X − 1 A) n = 0 o r lim n → ∞ (I − A X − 1) n = 0 (\ displaystyle \ lim _ (n \ to \ infty) (\ mathbf (I) - \ mathbf (X) ^ (- 1) \ mathbf (A)) ^ (n) = 0 \ mathrm (~ ~ or ~ ~) \ lim _ (n \ to \ infty) (\ mathbf (I) - \ mathbf (A) \ mathbf (X) ^ (- 1)) ^ (n) = 0) </Dd> </Dl>

When is a matrix said to be invertible