The most wireless communication systems uses the equalization in order to remove the inter-symbol interference (ISI) due to the multipath fading channel impairment. When the minimum mean square error (MMSE) equalizer is applied, the linear system solution requires very high computational complexity so that the equalization would be the most critical part with the decoder block. One of the effective algorithm for a linear system solution is Cholesky based method, which complexity is exponentially proportional to the length of the symbol sequence. However, the complexity of the Cholesky decomposition can be drastically reduced by using the properties of the channel correlation matrix, which is Hermitian, Toeplitz and banded. In this paper, the threshold value for the adaptive Cholesky algorithm is proposed and verified by the simulations using various channel parameters. The complexity reduction of the adaptive algorithm compared with the conventional algorithm without any significant bit error rate performance degradation.