  Given a real matrix A with n columns, the problem is to approximate the Gram
product AA^T by c << n weighted outer products of columns of A. Necessary and
sufficient conditions for the exact computation of AA^T (in exact arithmetic)
from c >= rank(A) columns depend on the right singular vector matrix of A. For
a Monte-Carlo matrix multiplication algorithm by Drineas et al. that samples
outer products, we present probabilistic bounds for the 2-norm relative error
due to randomization. The bounds depend on the stable rank or the rank of A,
but not on the matrix dimensions. Numerical experiments illustrate that the
bounds are informative, even for stringent success probabilities and matrices
of small dimension. We also derive bounds for the smallest singular value and
the condition number of matrices obtained by sampling rows from orthonormal
matrices.
