## Franklin's Notes

### Krylov method

A Krylov method is an iterative method which makes use of the Krylov spaces For instance, the Krylov spaces can be used to greatly simplify the process of solving the system $Ax=b$ for large matrices $A$. If we let $Q$ be the orthogonal projector onto $\mathcal{K}_n(A,b)$, and we search for an approximate solution $x=Q\tilde x$ to the system $Ax=b$ from the Krylov subspace $\mathcal K_n$, we can calculate it by solving the least squares problem $AQ\tilde x=b$, or solve the equation $Q^\ast AQ\tilde x = Q^\ast b$. But the matrix $Q^\ast A Q$ can be much smaller than $A$, if $n$ is much smaller than the size of $A$, which makes the problem much cheaper to solve.