We address the performance of airborne gamma detection systems equipped with a NaI(Tl) detector to monitor radionuclides in specific areas. In particular, we analyze the use of the fast singular value decomposition (FSVD) algorithm to improve the nuclide recognition ability of the system and effectively trace radioactivity in a complex background environment. We first present a theoretical analysis of the FSVD algorithm and illustrate the nuclide recognition algorithm step by step. The core of the algorithm is singular value decomposition and parameter estimation based on a Gaussian Markov linear regression model. From the estimated values of the parameters, information about radionuclides can be effectively extracted. We assume the presence of a strong background due to a high concentration of 222Rn and its progeny, which is simulated using GEANT4. By adding trace elements of 131I and 137Cs and changing the relative emissivity, the ratio of the total energy peak count of 131I and 137Cs to the background environment interval count of the corresponding 222Rn and its progeny are controlled. Assuming a counting ratio equal to 0.005, the FSVD algorithm is still able to effectively discriminate the presence of a small number of nuclides, reflecting very excellent recognition ability. Finally, based on data from an airborne gamma detection system in a self-control radon chamber, the FSVD algorithm is employed to recognize the trace of 137Cs nuclides in a strong radon background. A DURRIDGE RAD7 radon measuring instrument is used to monitor the radon concentration in the radon chamber. The actual measurement results show that the FSVD algorithm can effectively detect 137Cs nuclides.