To address the limitations of traditional grey relational analysis models, which fail to measure interaction between factors and do not account for data noise, a grey interactional relational analysis model based on singular value decomposition (SVD) is proposed. By introducing an interactional matrix and using SVD, the key singular values can be determined using a contribution rate criterion. Subsequently, grey relational coefficients for both individual factors and interaction effects are constructed, leading to the grey relational degrees considering both individual and interaction effects. The proposed model satisfies the properties of normalization, symmetry, and invariance under scalar transformation, while also overcoming the impact of the object arrangement order on relational degree. Finally, the model is used to analyze single-factor and interaction-driven influences on carbon emissions in the Yellow River Basin, identifying key influencing factors. The results demonstrate the rationality and effectiveness of the proposed model. Stability and permutation tests are further conducted to validate the robustness of the model.