Matrix polynomial expansion (MPE) based detector incurs either complicated computation of polynomial coefficients or slow convergence in uplink large-scale multiple-input and multiple-output (LS-MIMO) systems. To solve these issues, an improved MPE (IMPE) detector is proposed, which can speed up the convergence significantly with uncomplicated polynomial coefficients. However, a challenging problem of performing IMPE is needed to compute all the eigenvalues of channel covariance matrix in real time. Unfortunately, directly calculating the eigenvalues of the channel covariance matrix requires complexity, which is as costly as the matrix inverse. To this end, an inverse-transform-sampling based IMPE (ITS-IMPE) detector is proposed to enhance the convergence rate and accuracy in a simple way. First, the closed-form expression of the eigenvalue spectral cumulative distribution function of the channel covariance matrix is deduced analytically, which is a critical factor that influence the eigenvalues estimation. Second, the improved polynomial coefficients of ITS-IMPE are then introduced by a fast online ITS-based eigenvalues estimation algorithm and a least-squares fitting procedure, which achieve a well trade-off between precision and computation. Simulation results exhibit that ITS-IMPE detector is able to achieve a significant enhancement performance with much lower complexity compared with many reported detectors under Rayleigh fading channel and low spatial correlated channel.