A novel matrix multiplication framework for modeling genotype-by-environment interaction in genomic prediction

Accurate modeling of genotype-by-environment (G × E) interaction is a central challenge in plant breeding, as it determines how genetic potential is expressed across heterogeneous and changing environments. In genomic prediction, a widely adopted strategy represents G × E interaction through the Hadamard (element-wise) product of genotype and environment covariance matrices, capturing interaction effects via joint similarity. While effective, this formulation may not fully represent the structured complexity of G × E relationships observed in multi-environment trials. In this study, we extend a recently proposed matrix-product framework for covariance construction to explicitly model G × E interaction. Unlike the Hadamard approach, which captures similarity-by-similarity effects, the proposed method derives symmetric kernels from the matrix multiplication of genotype and environment kernels. These kernels retain complementary subsets of matrix-product information, but the numerical upper/lower assignment is tied to record ordering and should not be interpreted as an intrinsic biological direction. Because the direct matrix product and its symmetrized subcomponents are not guaranteed to be valid covariance matrices, positive semidefiniteness must be examined explicitly before the kernels are used as random-effect covariance structures. The proposed framework was evaluated using eleven multi-environment datasets in wheat and rice, comprising diverse traits and experimental structures. On average across datasets, models containing matrix-product-derived kernels showed higher predictive performance than the conventional Hadamard-only formulation, with an average COR difference of up to 13.2% for the combined model. These empirical gains should be interpreted as average predictive differences under the validation scheme used here rather than as a claim of universal or statistically significant superiority. These results show that matrix-product-derived terms can carry predictive information complementary to the conventional Hadamard interaction term. At the same time, the present revision makes explicit two structural limitations that must be considered when interpreting these kernels: positional triangular extraction is record-order dependent, and balanced incidence combined with centered genomic relationships can make the matrix product degenerate or nearly zero. These findings define both the practical scope of the current formulation and priorities for further methodological development.