  In this work, we present a method to exponentiate non-sparse indefinite
low-rank matrices on a quantum computer. Given an operation for accessing the
elements of the matrix, our method allows singular values and associated
singular vectors to be found quantum mechanically in a time exponentially
faster in the dimension of the matrix than known classical algorithms. The
method extends to non-Hermitian and non-square matrices via embedding matrices.
In the context of the generic singular value decomposition of a matrix, we
discuss the Procrustes problem of finding a closest isometry to a given matrix.
