  Most existing methodologies of estimating low-rank matrices rely on
Burer-Monteiro factorization, but these approaches can suffer from slow
convergence, especially when dealing with solutions characterized by a large
condition number, defined by the ratio of the largest to the $r$-th singular
values, where $r$ is the search rank. While methods such as Scaled Gradient
Descent have been proposed to address this issue, such methods are more
complicated and sometimes have weaker theoretical guarantees, for example, in
the rank-deficient setting. In contrast, this paper demonstrates the
effectiveness of the projected gradient descent algorithm. Firstly, its local
convergence rate is independent of the condition number. Secondly, under
conditions where the objective function is rank-$2r$ restricted $L$-smooth and
$\mu$-strongly convex, with $L/\mu < 3$, projected gradient descent with
appropriate step size converges linearly to the solution. Moreover, a perturbed
version of this algorithm effectively navigates away from saddle points,
converging to an approximate solution or a second-order local minimizer across
a wide range of step sizes. Furthermore, we establish that there are no
spurious local minimizers in estimating asymmetric low-rank matrices when the
objective function satisfies $L/\mu<3.$
