For the low rank approximation of time-dependent data matrices
and of solutions to matrix differential equations,
computational approach is proposed and analyzed.
In this method, the derivative is projected onto the tangent
space of the manifold of rank-r matrices at the current
approximation. With an appropriate decomposition of
rank-r matrices and their tangent matrices,
this yields nonlinear differential equations
that are well-suited for numerical integration.
The error analysis
compares the result with the pointwise best approximation in the Frobenius norm.
It is shown that the approach gives
locally quasi-optimal low rank approximations.
Numerical experiments illustrate the theoretical results.