A new method of extracting the relaxation matrix directly from absorption spectral data is formulated and applied to R-branch line mixing in HCN. The formulation makes use of a general iterative inversion algorithm based upon first-order sensitivity analysis and Tikhonov regularization. The recovered relaxation matrices describe line mixing much better than those derived from the fitting laws currently in use, and the inversion algorithm usually converges within just three iterations. This formulation presents the first known method for extracting the imaginary, off-diagonal elements of the relaxation matrix.