Krylov subspace
In linear algebra, the order-r Krylov subspace generated by an n-by-n matrix A and a vector b of dimension n is the linear subspace spanned by the images of b under the first r powers of A (starting from A 0 = I {\displaystyle A^{0}=I} ), that is, K r ( A , b ) = span { b , A b , A 2 b , … , A r − 1 b } . {\displaystyle {\mathcal {K}}_{r}(A,b)=\operatorname {span} \,\{b,Ab,A^{2}b,\ldots ,A^{r-1}b\}.} == Background == The concept is named after Russian applied mathematician and naval engineer Alexei Krylov, who published a paper about the concept in 1931.