The affine matrix rank minimization (AMRM) problem is to find a matrix of minimum rank that satisfies a given linear system constraint. It has many applications in some important areas such as control, recommender systems, matrix completion and network localization... However, the problem (AMRM) is NP-hard in general due to the combinational nature of the matrix rank function. There are many alternative functions have been proposed to substitute the matrix rank function, which lead to many corresponding alternative minimization problems solved efficiently by some popular convex or nonconvex optimization algorithms. In this paper, we propose a new nonconvex function, namely, $TL_{\alpha}^{\epsilon}$ function (with $0\leq\alpha<1$ and $\epsilon>0$), to approximate the rank function, and translate the NP-hard problem (AMRM) into the $TL_{p}^{\epsilon}$ function affine matrix rank minimization (TLAMRM) problem. Firstly, we study the equivalence of problem (AMRM) and (TLAMRM), and proved that the uniqueness of global minimizer of the problem (TLAMRM) also solves the NP-hard problem (AMRM) if the linear map $\mathcal{A}$ satisfies a restricted isometry property (RIP). Secondly, an iterative thresholding algorithm is proposed to solve the regularization problem (RTLAMRM) for all $0\leq\alpha<1$ and $\epsilon>0$. At last, some numerical results on low-rank matrix completion problems illustrated that our algorithm is able to recover a low-rank matrix, and the extensive numerical on image inpainting problems shown that our algorithm performs the best in finding a low-rank image compared with some state-of-art methods. (read more)