Approximations

$B$ (of rank

$j$) of a given large non-square matrix

$A$ are studied. Their optimal form is defined as a minimizer of the Frobenius norm of the difference

$A-B$. In principle, it can be constructed via a

$j$-dimensional part (projection) of a certain diagonalized form of

$A$. In such a setting one would need the full singular value decomposition of

$A$ (which generalizes the current diagonalization using two orthogonal matrices

$P$ and

$Q$). The paper offers a cheaper algorithm based on the Lanczos bidiagonalization. The authors perform the error and stopping analysis and demonstrate that a mere one-sided re-orthogonalization of this process guarantees a sufficient precision. A collection of matrices from several application areas is used in illustrative tests.