  A random matrix is likely to be well conditioned, and motivated by this well
known property we employ random matrix multipliers to advance some fundamental
matrix computations. This includes numerical stabilization of Gaussian
elimination with no pivoting as well as block Gaussian elimination,
approximation of the leading and trailing singular spaces of an ill conditioned
matrix, associated with its largest and smallest singular values, respectively,
and approximation of this matrix by low-rank matrices, with further extensions
to computing numerical ranks and the approximation of tensor decomposition. We
formally support the efficiency of the proposed techniques where we employ
Gaussian random multipliers, but our extensive tests have consistently produced
the same outcome where instead we used sparse and structured random
multipliers, defined by much fewer random parameters compared to the number of
their entries.
