  We study a type of Riemannian gradient descent (RGD) algorithm, designed
through Riemannian preconditioning, for optimization on $\mathcal{M}_k^{m\times
n}$ -- the set of $m\times n$ real matrices with a fixed rank $k$. Our analysis
is based on a quotient geometric view of $\mathcal{M}_k^{m\times n}$: by
identifying this set with the quotient manifold of a two-term product space
$\mathbb{R}_*^{m\times k}\times \mathbb{R}_*^{n\times k}$ of matrices with full
column rank via matrix factorization, we find an explicit form for the update
rule of the RGD algorithm, which leads to a novel approach to analysing their
convergence behavior in rank-constrained optimization. We then deduce some
interesting properties that reflect how RGD distinguishes from other matrix
factorization algorithms such as those based on the Euclidean geometry. In
particular, we show that the RGD algorithm are not only faster than Euclidean
gradient descent but also do not rely on the balancing technique to ensure its
efficiency while the latter does. Starting from the novel results, we further
show that this RGD algorithm is guaranteed to solve matrix sensing and matrix
completion problems with linear convergence rate, under mild conditions related
to the restricted positive definiteness property. Numerical experiments on
matrix sensing and completion are provided to demonstrate these properties.
