An eigendecomposition based method for two-dimensional frequency estimation is analyzed in this paper. This method, to be referred to as matrix pencil (MP) method, computes a smoothed data covariance matrix, then its eigendecomposition, and then the two-dimensional frequencies via a MP approach. The MP method is now known to be more efficient in computation than many other methods and able to provide a near optimum performance for relatively large signal-to-noise ratio (SNR). The aim of this paper is to provide a further analysis of the MP method assuming a moderate SNR. To make the problem tractable, a large two-dimensional data set is considered. In this paper, a number of fundamental relations inherent in the MP method are revealed which lead to a general expression of the large-sample covariances of the estimated two-dimensional frequencies. The large-sample covariances are reduced to a very simple form for the single two-dimensional frequency case. The theoretical covariances are verified by the simulation results.