  We introduce a definition of the volume for a general rectangular matrix,
which for square matrices is equivalent to the absolute value of the
determinant. We generalize results for square maximum-volume submatrices to the
case of rectangular maximal-volume submatrices, show connection of the
rectangular volume with optimal experimental design and provide estimates for
the growth of the coefficients and approximation error in spectral and
Chebyshev norms. Three promising applications of such submatrices are
presented: recommender systems, finding maximal elements in low-rank matrices
and preconditioning of overdetermined linear systems. The code is available
online.
