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 $A^{0}=I$ ), that is,^[1]^[2]

{\mathcal {K}}_{r}(A,b)=\operatorname {span} \,\{b,Ab,A^{2}b,\ldots ,A^{r-1}b\}.

^ Nocedal, Jorge; Wright, Stephen J. (2006). Numerical optimization. Springer series in operation research and financial engineering (2nd ed.). New York, NY: Springer. p. 108. ISBN 978-0-387-30303-1.
^ Simoncini, Valeria (2015), "Krylov Subspaces", in Nicholas J. Higham; et al. (eds.), The Princeton Companion to Applied Mathematics, Princeton University Press, pp. 113–114

[1] Nocedal, Jorge; Wright, Stephen J. (2006). Numerical optimization. Springer series in operation research and financial engineering (2nd ed.). New York, NY: Springer. p. 108. ISBN 978-0-387-30303-1.

[PrincetonCompanion-2] Simoncini, Valeria (2015), "Krylov Subspaces", in Nicholas J. Higham; et al. (eds.), The Princeton Companion to Applied Mathematics, Princeton University Press, pp. 113–114

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