New Results on Randomized Matrix Computations

The aim of this thesis is to present new results in randomized matrix computations. Specifically, and ultimately, we show how to modify, or preprocess an ill conditioned matrix having small numerical nullity (co-rank) into a nonsingular well conditioned matrix. This has intrinsic theoretical interest and we show a sample application to accurate solutions of nonsingular and ill conditioned linear systems. We discuss both multiplicative and additive preprocessing; in fact the multiplicative case assists in the derivation of the additive case. In the additive case, we approximate a nonsingular ill conditioned matrix by a singular well conditioned matrix which is then preprocessed into a nonsingular well conditioned matrix, which will also assist in the aforementioned ultimate goal