Motivated by recent work on two dimensional (2D) harmonic component recovery via atomic norm minimization (ANM), a fast 2D direction of arrival (DOA) off-grid estimation based on ANM method was proposed. By introducing a matrix atomic norm the 2D DOA estimation problem is turned into matrix atomic norm minimization (MANM) problem. Since the 2D-ANM gridless DOA estimation is processed by vectorizing the 2D into 1D estimation and solved via semi-definite programming (SDP), which is with high computational cost in 2D processing when the number of antennas increases to large size. In order to overcome this difficulty, a detail formulation of MANM problem via SDP method is offered in this paper, the MANM method converts the original $MN+1$ dimensions problem into a $M+N$ dimensions SDP problem and greatly reduces the computational complexity. In this paper we study the problem of 2D line spectrally-sparse signal recovery from partial noiseless observations and full noisy observations, both of which can be solved efficiently via MANM method and obtain high accuracy estimation of the true 2D angles. We give a sufficient condition of the optimality condition of the proposed method and prove an up bound of the expected error rate for denoising. Finally, numerical simulations are conducted to show the efficiency and performance of the proposed method, with comparisons against several existed sparse methods.