  Among randomized numerical linear algebra strategies, so-called sketching
procedures are emerging as effective reduction means to accelerate the
computation of Krylov subspace methods for, e.g., the solution of linear
systems, eigenvalue computations, and the approximation of matrix functions.
While there is plenty of experimental evidence showing that sketched Krylov
solvers may dramatically improve performance over standard Krylov methods, many
features of these schemes are still unexplored. We derive a new sketched
Arnoldi-type relation that allows us to obtain several different new
theoretical results. These lead to an improvement of our understanding of
sketched Krylov methods, in particular by explaining why the frequently
occurring sketched Ritz values far outside the spectral region of A do not
negatively influence the convergence of sketched Krylov methods for f (A)b. Our
findings also help to identify, among several possible equivalent formulations,
the most suitable sketched approximations according to their numerical
stability properties. These results are also employed to analyze the error of
sketched Krylov methods in the approximation of the action of matrix functions,
significantly contributing to the theory available in the current literature.
