For the low rank approximation of time-dependent data matrices
and of solutions to matrix differential equations,
an increment-based
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.