  We take a random matrix theory approach to random sketching and show an
asymptotic first-order equivalence of the regularized sketched pseudoinverse of
a positive semidefinite matrix to a certain evaluation of the resolvent of the
same matrix. We focus on real-valued regularization and extend previous results
on an asymptotic equivalence of random matrices to the real setting, providing
a precise characterization of the equivalence even under negative
regularization, including a precise characterization of the smallest nonzero
eigenvalue of the sketched matrix, which may be of independent interest. We
then further characterize the second-order equivalence of the sketched
pseudoinverse. We also apply our results to the analysis of the
sketch-and-project method and to sketched ridge regression. Lastly, we prove
that these results generalize to asymptotically free sketching matrices,
obtaining the resulting equivalence for orthogonal sketching matrices and
comparing our results to several common sketches used in practice.
